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diff --git a/ports/sysdeps/ia64/fpu/s_tanl.S b/ports/sysdeps/ia64/fpu/s_tanl.S new file mode 100644 index 0000000000..b59936c4a1 --- /dev/null +++ b/ports/sysdeps/ia64/fpu/s_tanl.S @@ -0,0 +1,3248 @@ +.file "tancotl.s" + + +// Copyright (c) 2000 - 2004, Intel Corporation +// All rights reserved. +// +// Contributed 2000 by the Intel Numerics Group, Intel Corporation +// +// Redistribution and use in source and binary forms, with or without +// modification, are permitted provided that the following conditions are +// met: +// +// * Redistributions of source code must retain the above copyright +// notice, this list of conditions and the following disclaimer. +// +// * Redistributions in binary form must reproduce the above copyright +// notice, this list of conditions and the following disclaimer in the +// documentation and/or other materials provided with the distribution. +// +// * The name of Intel Corporation may not be used to endorse or promote +// products derived from this software without specific prior written +// permission. + +// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS +// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY +// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING +// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +// +// Intel Corporation is the author of this code, and requests that all +// problem reports or change requests be submitted to it directly at +// http://www.intel.com/software/products/opensource/libraries/num.htm. +// +//********************************************************************* +// +// History: +// +// 02/02/00 (hand-optimized) +// 04/04/00 Unwind support added +// 12/28/00 Fixed false invalid flags +// 02/06/02 Improved speed +// 05/07/02 Changed interface to __libm_pi_by_2_reduce +// 05/30/02 Added cotl +// 02/10/03 Reordered header: .section, .global, .proc, .align; +// used data8 for long double table values +// 05/15/03 Reformatted data tables +// 10/26/04 Avoided using r14-31 as scratch so not clobbered by dynamic loader +// +//********************************************************************* +// +// Functions: tanl(x) = tangent(x), for double-extended precision x values +// cotl(x) = cotangent(x), for double-extended precision x values +// +//********************************************************************* +// +// Resources Used: +// +// Floating-Point Registers: f8 (Input and Return Value) +// f9-f15 +// f32-f121 +// +// General Purpose Registers: +// r32-r70 +// +// Predicate Registers: p6-p15 +// +//********************************************************************* +// +// IEEE Special Conditions for tanl: +// +// Denormal fault raised on denormal inputs +// Overflow exceptions do not occur +// Underflow exceptions raised when appropriate for tan +// (No specialized error handling for this routine) +// Inexact raised when appropriate by algorithm +// +// tanl(SNaN) = QNaN +// tanl(QNaN) = QNaN +// tanl(inf) = QNaN +// tanl(+/-0) = +/-0 +// +//********************************************************************* +// +// IEEE Special Conditions for cotl: +// +// Denormal fault raised on denormal inputs +// Overflow exceptions occur at zero and near zero +// Underflow exceptions do not occur +// Inexact raised when appropriate by algorithm +// +// cotl(SNaN) = QNaN +// cotl(QNaN) = QNaN +// cotl(inf) = QNaN +// cotl(+/-0) = +/-Inf and error handling is called +// +//********************************************************************* +// +// Below are mathematical and algorithmic descriptions for tanl. +// For cotl we use next identity cot(x) = -tan(x + Pi/2). +// So, to compute cot(x) we just need to increment N (N = N + 1) +// and invert sign of the computed result. +// +//********************************************************************* +// +// Mathematical Description +// +// We consider the computation of FPTANL of Arg. Now, given +// +// Arg = N pi/2 + alpha, |alpha| <= pi/4, +// +// basic mathematical relationship shows that +// +// tan( Arg ) = tan( alpha ) if N is even; +// = -cot( alpha ) otherwise. +// +// The value of alpha is obtained by argument reduction and +// represented by two working precision numbers r and c where +// +// alpha = r + c accurately. +// +// The reduction method is described in a previous write up. +// The argument reduction scheme identifies 4 cases. For Cases 2 +// and 4, because |alpha| is small, tan(r+c) and -cot(r+c) can be +// computed very easily by 2 or 3 terms of the Taylor series +// expansion as follows: +// +// Case 2: +// ------- +// +// tan(r + c) = r + c + r^3/3 ...accurately +// -cot(r + c) = -1/(r+c) + r/3 ...accurately +// +// Case 4: +// ------- +// +// tan(r + c) = r + c + r^3/3 + 2r^5/15 ...accurately +// -cot(r + c) = -1/(r+c) + r/3 + r^3/45 ...accurately +// +// +// The only cases left are Cases 1 and 3 of the argument reduction +// procedure. These two cases will be merged since after the +// argument is reduced in either cases, we have the reduced argument +// represented as r + c and that the magnitude |r + c| is not small +// enough to allow the usage of a very short approximation. +// +// The greatest challenge of this task is that the second terms of +// the Taylor series for tan(r) and -cot(r) +// +// r + r^3/3 + 2 r^5/15 + ... +// +// and +// +// -1/r + r/3 + r^3/45 + ... +// +// are not very small when |r| is close to pi/4 and the rounding +// errors will be a concern if simple polynomial accumulation is +// used. When |r| < 2^(-2), however, the second terms will be small +// enough (5 bits or so of right shift) that a normal Horner +// recurrence suffices. Hence there are two cases that we consider +// in the accurate computation of tan(r) and cot(r), |r| <= pi/4. +// +// Case small_r: |r| < 2^(-2) +// -------------------------- +// +// Since Arg = N pi/4 + r + c accurately, we have +// +// tan(Arg) = tan(r+c) for N even, +// = -cot(r+c) otherwise. +// +// Here for this case, both tan(r) and -cot(r) can be approximated +// by simple polynomials: +// +// tan(r) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19 +// -cot(r) = -1/r + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13 +// +// accurately. Since |r| is relatively small, tan(r+c) and +// -cot(r+c) can be accurately approximated by replacing r with +// r+c only in the first two terms of the corresponding polynomials. +// +// Note that P1_1 (and Q1_1 for that matter) approximates 1/3 to +// almost 64 sig. bits, thus +// +// P1_1 (r+c)^3 = P1_1 r^3 + c * r^2 accurately. +// +// Hence, +// +// tan(r+c) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19 +// + c*(1 + r^2) +// +// -cot(r+c) = -1/(r+c) + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13 +// + Q1_1*c +// +// +// Case normal_r: 2^(-2) <= |r| <= pi/4 +// ------------------------------------ +// +// This case is more likely than the previous one if one considers +// r to be uniformly distributed in [-pi/4 pi/4]. +// +// The required calculation is either +// +// tan(r + c) = tan(r) + correction, or +// -cot(r + c) = -cot(r) + correction. +// +// Specifically, +// +// tan(r + c) = tan(r) + c tan'(r) + O(c^2) +// = tan(r) + c sec^2(r) + O(c^2) +// = tan(r) + c SEC_sq ...accurately +// as long as SEC_sq approximates sec^2(r) +// to, say, 5 bits or so. +// +// Similarly, +// +// -cot(r + c) = -cot(r) - c cot'(r) + O(c^2) +// = -cot(r) + c csc^2(r) + O(c^2) +// = -cot(r) + c CSC_sq ...accurately +// as long as CSC_sq approximates csc^2(r) +// to, say, 5 bits or so. +// +// We therefore concentrate on accurately calculating tan(r) and +// cot(r) for a working-precision number r, |r| <= pi/4 to within +// 0.1% or so. +// +// We will employ a table-driven approach. Let +// +// r = sgn_r * 2^k * 1.b_1 b_2 ... b_5 ... b_63 +// = sgn_r * ( B + x ) +// +// where +// +// B = 2^k * 1.b_1 b_2 ... b_5 1 +// x = |r| - B +// +// Now, +// tan(B) + tan(x) +// tan( B + x ) = ------------------------ +// 1 - tan(B)*tan(x) +// +// / \ +// | tan(B) + tan(x) | + +// = tan(B) + | ------------------------ - tan(B) | +// | 1 - tan(B)*tan(x) | +// \ / +// +// sec^2(B) * tan(x) +// = tan(B) + ------------------------ +// 1 - tan(B)*tan(x) +// +// (1/[sin(B)*cos(B)]) * tan(x) +// = tan(B) + -------------------------------- +// cot(B) - tan(x) +// +// +// Clearly, the values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are +// calculated beforehand and stored in a table. Since +// +// |x| <= 2^k * 2^(-6) <= 2^(-7) (because k = -1, -2) +// +// a very short polynomial will be sufficient to approximate tan(x) +// accurately. The details involved in computing the last expression +// will be given in the next section on algorithm description. +// +// +// Now, we turn to the case where cot( B + x ) is needed. +// +// +// 1 - tan(B)*tan(x) +// cot( B + x ) = ------------------------ +// tan(B) + tan(x) +// +// / \ +// | 1 - tan(B)*tan(x) | + +// = cot(B) + | ----------------------- - cot(B) | +// | tan(B) + tan(x) | +// \ / +// +// [tan(B) + cot(B)] * tan(x) +// = cot(B) - ---------------------------- +// tan(B) + tan(x) +// +// (1/[sin(B)*cos(B)]) * tan(x) +// = cot(B) - -------------------------------- +// tan(B) + tan(x) +// +// +// Note that the values of tan(B), cot(B) and 1/(sin(B)*cos(B)) that +// are needed are the same set of values needed in the previous +// case. +// +// Finally, we can put all the ingredients together as follows: +// +// Arg = N * pi/2 + r + c ...accurately +// +// tan(Arg) = tan(r) + correction if N is even; +// = -cot(r) + correction otherwise. +// +// For Cases 2 and 4, +// +// Case 2: +// tan(Arg) = tan(r + c) = r + c + r^3/3 N even +// = -cot(r + c) = -1/(r+c) + r/3 N odd +// Case 4: +// tan(Arg) = tan(r + c) = r + c + r^3/3 + 2r^5/15 N even +// = -cot(r + c) = -1/(r+c) + r/3 + r^3/45 N odd +// +// +// For Cases 1 and 3, +// +// Case small_r: |r| < 2^(-2) +// +// tan(Arg) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19 +// + c*(1 + r^2) N even +// +// = -1/(r+c) + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13 +// + Q1_1*c N odd +// +// Case normal_r: 2^(-2) <= |r| <= pi/4 +// +// tan(Arg) = tan(r) + c * sec^2(r) N even +// = -cot(r) + c * csc^2(r) otherwise +// +// For N even, +// +// tan(Arg) = tan(r) + c*sec^2(r) +// = tan( sgn_r * (B+x) ) + c * sec^2(|r|) +// = sgn_r * ( tan(B+x) + sgn_r*c*sec^2(|r|) ) +// = sgn_r * ( tan(B+x) + sgn_r*c*sec^2(B) ) +// +// since B approximates |r| to 2^(-6) in relative accuracy. +// +// / (1/[sin(B)*cos(B)]) * tan(x) +// tan(Arg) = sgn_r * | tan(B) + -------------------------------- +// \ cot(B) - tan(x) +// \ +// + CORR | + +// / +// where +// +// CORR = sgn_r*c*tan(B)*SC_inv(B); SC_inv(B) = 1/(sin(B)*cos(B)). +// +// For N odd, +// +// tan(Arg) = -cot(r) + c*csc^2(r) +// = -cot( sgn_r * (B+x) ) + c * csc^2(|r|) +// = sgn_r * ( -cot(B+x) + sgn_r*c*csc^2(|r|) ) +// = sgn_r * ( -cot(B+x) + sgn_r*c*csc^2(B) ) +// +// since B approximates |r| to 2^(-6) in relative accuracy. +// +// / (1/[sin(B)*cos(B)]) * tan(x) +// tan(Arg) = sgn_r * | -cot(B) + -------------------------------- +// \ tan(B) + tan(x) +// \ +// + CORR | + +// / +// where +// +// CORR = sgn_r*c*cot(B)*SC_inv(B); SC_inv(B) = 1/(sin(B)*cos(B)). +// +// +// The actual algorithm prescribes how all the mathematical formulas +// are calculated. +// +// +// 2. Algorithmic Description +// ========================== +// +// 2.1 Computation for Cases 2 and 4. +// ---------------------------------- +// +// For Case 2, we use two-term polynomials. +// +// For N even, +// +// rsq := r * r +// Poly := c + r * rsq * P1_1 +// Result := r + Poly ...in user-defined rounding +// +// For N odd, +// S_hi := -frcpa(r) ...8 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits +// S_lo := S_hi*( (1 + S_hi*r) + S_hi*c ) +// ...S_hi + S_lo is -1/(r+c) to extra precision +// S_lo := S_lo + Q1_1*r +// +// Result := S_hi + S_lo ...in user-defined rounding +// +// For Case 4, we use three-term polynomials +// +// For N even, +// +// rsq := r * r +// Poly := c + r * rsq * (P1_1 + rsq * P1_2) +// Result := r + Poly ...in user-defined rounding +// +// For N odd, +// S_hi := -frcpa(r) ...8 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits +// S_lo := S_hi*( (1 + S_hi*r) + S_hi*c ) +// ...S_hi + S_lo is -1/(r+c) to extra precision +// rsq := r * r +// P := Q1_1 + rsq*Q1_2 +// S_lo := S_lo + r*P +// +// Result := S_hi + S_lo ...in user-defined rounding +// +// +// Note that the coefficients P1_1, P1_2, Q1_1, and Q1_2 are +// the same as those used in the small_r case of Cases 1 and 3 +// below. +// +// +// 2.2 Computation for Cases 1 and 3. +// ---------------------------------- +// This is further divided into the case of small_r, +// where |r| < 2^(-2), and the case of normal_r, where |r| lies between +// 2^(-2) and pi/4. +// +// Algorithm for the case of small_r +// --------------------------------- +// +// For N even, +// rsq := r * r +// Poly1 := rsq*(P1_1 + rsq*(P1_2 + rsq*P1_3)) +// r_to_the_8 := rsq * rsq +// r_to_the_8 := r_to_the_8 * r_to_the_8 +// Poly2 := P1_4 + rsq*(P1_5 + rsq*(P1_6 + ... rsq*P1_9)) +// CORR := c * ( 1 + rsq ) +// Poly := Poly1 + r_to_the_8*Poly2 +// Poly := r*Poly + CORR +// Result := r + Poly ...in user-defined rounding +// ...note that Poly1 and r_to_the_8 can be computed in parallel +// ...with Poly2 (Poly1 is intentionally set to be much +// ...shorter than Poly2 so that r_to_the_8 and CORR can be hidden) +// +// For N odd, +// S_hi := -frcpa(r) ...8 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits +// S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits +// S_lo := S_hi*( (1 + S_hi*r) + S_hi*c ) +// ...S_hi + S_lo is -1/(r+c) to extra precision +// S_lo := S_lo + Q1_1*c +// +// ...S_hi and S_lo are computed in parallel with +// ...the following +// rsq := r*r +// P := Q1_1 + rsq*(Q1_2 + rsq*(Q1_3 + ... + rsq*Q1_7)) +// +// Poly := r*P + S_lo +// Result := S_hi + Poly ...in user-defined rounding +// +// +// Algorithm for the case of normal_r +// ---------------------------------- +// +// Here, we first consider the computation of tan( r + c ). As +// presented in the previous section, +// +// tan( r + c ) = tan(r) + c * sec^2(r) +// = sgn_r * [ tan(B+x) + CORR ] +// CORR = sgn_r * c * tan(B) * 1/[sin(B)*cos(B)] +// +// because sec^2(r) = sec^(|r|), and B approximate |r| to 6.5 bits. +// +// tan( r + c ) = +// / (1/[sin(B)*cos(B)]) * tan(x) +// sgn_r * | tan(B) + -------------------------------- + +// \ cot(B) - tan(x) +// \ +// CORR | + +// / +// +// The values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are +// calculated beforehand and stored in a table. Specifically, +// the table values are +// +// tan(B) as T_hi + T_lo; +// cot(B) as C_hi + C_lo; +// 1/[sin(B)*cos(B)] as SC_inv +// +// T_hi, C_hi are in double-precision memory format; +// T_lo, C_lo are in single-precision memory format; +// SC_inv is in extended-precision memory format. +// +// The value of tan(x) will be approximated by a short polynomial of +// the form +// +// tan(x) as x + x * P, where +// P = x^2 * (P2_1 + x^2 * (P2_2 + x^2 * P2_3)) +// +// Because |x| <= 2^(-7), cot(B) - x approximates cot(B) - tan(x) +// to a relative accuracy better than 2^(-20). Thus, a good +// initial guess of 1/( cot(B) - tan(x) ) to initiate the iterative +// division is: +// +// 1/(cot(B) - tan(x)) is approximately +// 1/(cot(B) - x) is +// tan(B)/(1 - x*tan(B)) is approximately +// T_hi / ( 1 - T_hi * x ) is approximately +// +// T_hi * [ 1 + (Thi * x) + (T_hi * x)^2 ] +// +// The calculation of tan(r+c) therefore proceed as follows: +// +// Tx := T_hi * x +// xsq := x * x +// +// V_hi := T_hi*(1 + Tx*(1 + Tx)) +// P := xsq * (P1_1 + xsq*(P1_2 + xsq*P1_3)) +// ...V_hi serves as an initial guess of 1/(cot(B) - tan(x)) +// ...good to about 20 bits of accuracy +// +// tanx := x + x*P +// D := C_hi - tanx +// ...D is a double precision denominator: cot(B) - tan(x) +// +// V_hi := V_hi + V_hi*(1 - V_hi*D) +// ....V_hi approximates 1/(cot(B)-tan(x)) to 40 bits +// +// V_lo := V_hi * ( [ (1 - V_hi*C_hi) + V_hi*tanx ] +// - V_hi*C_lo ) ...observe all order +// ...V_hi + V_lo approximates 1/(cot(B) - tan(x)) +// ...to extra accuracy +// +// ... SC_inv(B) * (x + x*P) +// ... tan(B) + ------------------------- + CORR +// ... cot(B) - (x + x*P) +// ... +// ... = tan(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR +// ... +// +// Sx := SC_inv * x +// CORR := sgn_r * c * SC_inv * T_hi +// +// ...put the ingredients together to compute +// ... SC_inv(B) * (x + x*P) +// ... tan(B) + ------------------------- + CORR +// ... cot(B) - (x + x*P) +// ... +// ... = tan(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR +// ... +// ... = T_hi + T_lo + CORR + +// ... Sx * V_hi + Sx * V_lo + Sx * P *(V_hi + V_lo) +// +// CORR := CORR + T_lo +// tail := V_lo + P*(V_hi + V_lo) +// tail := Sx * tail + CORR +// tail := Sx * V_hi + tail +// T_hi := sgn_r * T_hi +// +// ...T_hi + sgn_r*tail now approximate +// ...sgn_r*(tan(B+x) + CORR) accurately +// +// Result := T_hi + sgn_r*tail ...in user-defined +// ...rounding control +// ...It is crucial that independent paths be fully +// ...exploited for performance's sake. +// +// +// Next, we consider the computation of -cot( r + c ). As +// presented in the previous section, +// +// -cot( r + c ) = -cot(r) + c * csc^2(r) +// = sgn_r * [ -cot(B+x) + CORR ] +// CORR = sgn_r * c * cot(B) * 1/[sin(B)*cos(B)] +// +// because csc^2(r) = csc^(|r|), and B approximate |r| to 6.5 bits. +// +// -cot( r + c ) = +// / (1/[sin(B)*cos(B)]) * tan(x) +// sgn_r * | -cot(B) + -------------------------------- + +// \ tan(B) + tan(x) +// \ +// CORR | + +// / +// +// The values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are +// calculated beforehand and stored in a table. Specifically, +// the table values are +// +// tan(B) as T_hi + T_lo; +// cot(B) as C_hi + C_lo; +// 1/[sin(B)*cos(B)] as SC_inv +// +// T_hi, C_hi are in double-precision memory format; +// T_lo, C_lo are in single-precision memory format; +// SC_inv is in extended-precision memory format. +// +// The value of tan(x) will be approximated by a short polynomial of +// the form +// +// tan(x) as x + x * P, where +// P = x^2 * (P2_1 + x^2 * (P2_2 + x^2 * P2_3)) +// +// Because |x| <= 2^(-7), tan(B) + x approximates tan(B) + tan(x) +// to a relative accuracy better than 2^(-18). Thus, a good +// initial guess of 1/( tan(B) + tan(x) ) to initiate the iterative +// division is: +// +// 1/(tan(B) + tan(x)) is approximately +// 1/(tan(B) + x) is +// cot(B)/(1 + x*cot(B)) is approximately +// C_hi / ( 1 + C_hi * x ) is approximately +// +// C_hi * [ 1 - (C_hi * x) + (C_hi * x)^2 ] +// +// The calculation of -cot(r+c) therefore proceed as follows: +// +// Cx := C_hi * x +// xsq := x * x +// +// V_hi := C_hi*(1 - Cx*(1 - Cx)) +// P := xsq * (P1_1 + xsq*(P1_2 + xsq*P1_3)) +// ...V_hi serves as an initial guess of 1/(tan(B) + tan(x)) +// ...good to about 18 bits of accuracy +// +// tanx := x + x*P +// D := T_hi + tanx +// ...D is a double precision denominator: tan(B) + tan(x) +// +// V_hi := V_hi + V_hi*(1 - V_hi*D) +// ....V_hi approximates 1/(tan(B)+tan(x)) to 40 bits +// +// V_lo := V_hi * ( [ (1 - V_hi*T_hi) - V_hi*tanx ] +// - V_hi*T_lo ) ...observe all order +// ...V_hi + V_lo approximates 1/(tan(B) + tan(x)) +// ...to extra accuracy +// +// ... SC_inv(B) * (x + x*P) +// ... -cot(B) + ------------------------- + CORR +// ... tan(B) + (x + x*P) +// ... +// ... =-cot(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR +// ... +// +// Sx := SC_inv * x +// CORR := sgn_r * c * SC_inv * C_hi +// +// ...put the ingredients together to compute +// ... SC_inv(B) * (x + x*P) +// ... -cot(B) + ------------------------- + CORR +// ... tan(B) + (x + x*P) +// ... +// ... =-cot(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR +// ... +// ... =-C_hi - C_lo + CORR + +// ... Sx * V_hi + Sx * V_lo + Sx * P *(V_hi + V_lo) +// +// CORR := CORR - C_lo +// tail := V_lo + P*(V_hi + V_lo) +// tail := Sx * tail + CORR +// tail := Sx * V_hi + tail +// C_hi := -sgn_r * C_hi +// +// ...C_hi + sgn_r*tail now approximates +// ...sgn_r*(-cot(B+x) + CORR) accurately +// +// Result := C_hi + sgn_r*tail in user-defined rounding control +// ...It is crucial that independent paths be fully +// ...exploited for performance's sake. +// +// 3. Implementation Notes +// ======================= +// +// Table entries T_hi, T_lo; C_hi, C_lo; SC_inv +// +// Recall that 2^(-2) <= |r| <= pi/4; +// +// r = sgn_r * 2^k * 1.b_1 b_2 ... b_63 +// +// and +// +// B = 2^k * 1.b_1 b_2 b_3 b_4 b_5 1 +// +// Thus, for k = -2, possible values of B are +// +// B = 2^(-2) * ( 1 + index/32 + 1/64 ), +// index ranges from 0 to 31 +// +// For k = -1, however, since |r| <= pi/4 = 0.78... +// possible values of B are +// +// B = 2^(-1) * ( 1 + index/32 + 1/64 ) +// index ranges from 0 to 19. +// +// + +RODATA +.align 16 + +LOCAL_OBJECT_START(TANL_BASE_CONSTANTS) + +tanl_table_1: +data8 0xA2F9836E4E44152A, 0x00003FFE // two_by_pi +data8 0xC84D32B0CE81B9F1, 0x00004016 // P_0 +data8 0xC90FDAA22168C235, 0x00003FFF // P_1 +data8 0xECE675D1FC8F8CBB, 0x0000BFBD // P_2 +data8 0xB7ED8FBBACC19C60, 0x0000BF7C // P_3 +LOCAL_OBJECT_END(TANL_BASE_CONSTANTS) + +LOCAL_OBJECT_START(tanl_table_2) +data8 0xC90FDAA22168C234, 0x00003FFE // PI_BY_4 +data8 0xA397E5046EC6B45A, 0x00003FE7 // Inv_P_0 +data8 0x8D848E89DBD171A1, 0x0000BFBF // d_1 +data8 0xD5394C3618A66F8E, 0x0000BF7C // d_2 +data4 0x3E800000 // two**-2 +data4 0xBE800000 // -two**-2 +data4 0x00000000 // pad +data4 0x00000000 // pad +LOCAL_OBJECT_END(tanl_table_2) + +LOCAL_OBJECT_START(tanl_table_p1) +data8 0xAAAAAAAAAAAAAABD, 0x00003FFD // P1_1 +data8 0x8888888888882E6A, 0x00003FFC // P1_2 +data8 0xDD0DD0DD0F0177B6, 0x00003FFA // P1_3 +data8 0xB327A440646B8C6D, 0x00003FF9 // P1_4 +data8 0x91371B251D5F7D20, 0x00003FF8 // P1_5 +data8 0xEB69A5F161C67914, 0x00003FF6 // P1_6 +data8 0xBEDD37BE019318D2, 0x00003FF5 // P1_7 +data8 0x9979B1463C794015, 0x00003FF4 // P1_8 +data8 0x8EBD21A38C6EB58A, 0x00003FF3 // P1_9 +LOCAL_OBJECT_END(tanl_table_p1) + +LOCAL_OBJECT_START(tanl_table_q1) +data8 0xAAAAAAAAAAAAAAB4, 0x00003FFD // Q1_1 +data8 0xB60B60B60B5FC93E, 0x00003FF9 // Q1_2 +data8 0x8AB355E00C9BBFBF, 0x00003FF6 // Q1_3 +data8 0xDDEBBC89CBEE3D4C, 0x00003FF2 // Q1_4 +data8 0xB3548A685F80BBB6, 0x00003FEF // Q1_5 +data8 0x913625604CED5BF1, 0x00003FEC // Q1_6 +data8 0xF189D95A8EE92A83, 0x00003FE8 // Q1_7 +LOCAL_OBJECT_END(tanl_table_q1) + +LOCAL_OBJECT_START(tanl_table_p2) +data8 0xAAAAAAAAAAAB362F, 0x00003FFD // P2_1 +data8 0x88888886E97A6097, 0x00003FFC // P2_2 +data8 0xDD108EE025E716A1, 0x00003FFA // P2_3 +LOCAL_OBJECT_END(tanl_table_p2) + +LOCAL_OBJECT_START(tanl_table_tm2) +// +// Entries T_hi double-precision memory format +// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) +// Entries T_lo single-precision memory format +// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) +// +data8 0x3FD09BC362400794 +data4 0x23A05C32, 0x00000000 +data8 0x3FD124A9DFFBC074 +data4 0x240078B2, 0x00000000 +data8 0x3FD1AE235BD4920F +data4 0x23826B8E, 0x00000000 +data8 0x3FD2383515E2701D +data4 0x22D31154, 0x00000000 +data8 0x3FD2C2E463739C2D +data4 0x2265C9E2, 0x00000000 +data8 0x3FD34E36AFEEA48B +data4 0x245C05EB, 0x00000000 +data8 0x3FD3DA317DBB35D1 +data4 0x24749F2D, 0x00000000 +data8 0x3FD466DA67321619 +data4 0x2462CECE, 0x00000000 +data8 0x3FD4F4371F94A4D5 +data4 0x246D0DF1, 0x00000000 +data8 0x3FD5824D740C3E6D +data4 0x240A85B5, 0x00000000 +data8 0x3FD611234CB1E73D +data4 0x23F96E33, 0x00000000 +data8 0x3FD6A0BEAD9EA64B +data4 0x247C5393, 0x00000000 +data8 0x3FD73125B804FD01 +data4 0x241F3B29, 0x00000000 +data8 0x3FD7C25EAB53EE83 +data4 0x2479989B, 0x00000000 +data8 0x3FD8546FE6640EED +data4 0x23B343BC, 0x00000000 +data8 0x3FD8E75FE8AF1892 +data4 0x241454D1, 0x00000000 +data8 0x3FD97B3553928BDA +data4 0x238613D9, 0x00000000 +data8 0x3FDA0FF6EB9DE4DE +data4 0x22859FA7, 0x00000000 +data8 0x3FDAA5AB99ECF92D +data4 0x237A6D06, 0x00000000 +data8 0x3FDB3C5A6D8F1796 +data4 0x23952F6C, 0x00000000 +data8 0x3FDBD40A9CFB8BE4 +data4 0x2280FC95, 0x00000000 +data8 0x3FDC6CC387943100 +data4 0x245D2EC0, 0x00000000 +data8 0x3FDD068CB736C500 +data4 0x23C4AD7D, 0x00000000 +data8 0x3FDDA16DE1DDBC31 +data4 0x23D076E6, 0x00000000 +data8 0x3FDE3D6EEB515A93 +data4 0x244809A6, 0x00000000 +data8 0x3FDEDA97E6E9E5F1 +data4 0x220856C8, 0x00000000 +data8 0x3FDF78F11963CE69 +data4 0x244BE993, 0x00000000 +data8 0x3FE00C417D635BCE +data4 0x23D21799, 0x00000000 +data8 0x3FE05CAB1C302CD3 +data4 0x248A1B1D, 0x00000000 +data8 0x3FE0ADB9DB6A1FA0 +data4 0x23D53E33, 0x00000000 +data8 0x3FE0FF724A20BA81 +data4 0x24DB9ED5, 0x00000000 +data8 0x3FE151D9153FA6F5 +data4 0x24E9E451, 0x00000000 +LOCAL_OBJECT_END(tanl_table_tm2) + +LOCAL_OBJECT_START(tanl_table_tm1) +// +// Entries T_hi double-precision memory format +// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) +// Entries T_lo single-precision memory format +// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) +// +data8 0x3FE1CEC4BA1BE39E +data4 0x24B60F9E, 0x00000000 +data8 0x3FE277E45ABD9B2D +data4 0x248C2474, 0x00000000 +data8 0x3FE324180272B110 +data4 0x247B8311, 0x00000000 +data8 0x3FE3D38B890E2DF0 +data4 0x24C55751, 0x00000000 +data8 0x3FE4866D46236871 +data4 0x24E5BC34, 0x00000000 +data8 0x3FE53CEE45E044B0 +data4 0x24001BA4, 0x00000000 +data8 0x3FE5F74282EC06E4 +data4 0x24B973DC, 0x00000000 +data8 0x3FE6B5A125DF43F9 +data4 0x24895440, 0x00000000 +data8 0x3FE77844CAFD348C +data4 0x240021CA, 0x00000000 +data8 0x3FE83F6BCEED6B92 +data4 0x24C45372, 0x00000000 +data8 0x3FE90B58A34F3665 +data4 0x240DAD33, 0x00000000 +data8 0x3FE9DC522C1E56B4 +data4 0x24F846CE, 0x00000000 +data8 0x3FEAB2A427041578 +data4 0x2323FB6E, 0x00000000 +data8 0x3FEB8E9F9DD8C373 +data4 0x24B3090B, 0x00000000 +data8 0x3FEC709B65C9AA7B +data4 0x2449F611, 0x00000000 +data8 0x3FED58F4ACCF8435 +data4 0x23616A7E, 0x00000000 +data8 0x3FEE480F97635082 +data4 0x24C2FEAE, 0x00000000 +data8 0x3FEF3E57F0ACC544 +data4 0x242CE964, 0x00000000 +data8 0x3FF01E20F7E06E4B +data4 0x2480D3EE, 0x00000000 +data8 0x3FF0A1258A798A69 +data4 0x24DB8967, 0x00000000 +LOCAL_OBJECT_END(tanl_table_tm1) + +LOCAL_OBJECT_START(tanl_table_cm2) +// +// Entries C_hi double-precision memory format +// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) +// Entries C_lo single-precision memory format +// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) +// +data8 0x400ED3E2E63EFBD0 +data4 0x259D94D4, 0x00000000 +data8 0x400DDDB4C515DAB5 +data4 0x245F0537, 0x00000000 +data8 0x400CF57ABE19A79F +data4 0x25D4EA9F, 0x00000000 +data8 0x400C1A06D15298ED +data4 0x24AE40A0, 0x00000000 +data8 0x400B4A4C164B2708 +data4 0x25A5AAB6, 0x00000000 +data8 0x400A855A5285B068 +data4 0x25524F18, 0x00000000 +data8 0x4009CA5A3FFA549F +data4 0x24C999C0, 0x00000000 +data8 0x4009188A646AF623 +data4 0x254FD801, 0x00000000 +data8 0x40086F3C6084D0E7 +data4 0x2560F5FD, 0x00000000 +data8 0x4007CDD2A29A76EE +data4 0x255B9D19, 0x00000000 +data8 0x400733BE6C8ECA95 +data4 0x25CB021B, 0x00000000 +data8 0x4006A07E1F8DDC52 +data4 0x24AB4722, 0x00000000 +data8 0x4006139BC298AD58 +data4 0x252764E2, 0x00000000 +data8 0x40058CABBAD7164B +data4 0x24DAF5DB, 0x00000000 +data8 0x40050B4BAE31A5D3 +data4 0x25EA20F4, 0x00000000 +data8 0x40048F2189F85A8A +data4 0x2583A3E8, 0x00000000 +data8 0x400417DAA862380D +data4 0x25DCC4CC, 0x00000000 +data8 0x4003A52B1088FCFE +data4 0x2430A492, 0x00000000 +data8 0x400336CCCD3527D5 +data4 0x255F77CF, 0x00000000 +data8 0x4002CC7F5760766D +data4 0x25DA0BDA, 0x00000000 +data8 0x4002660711CE02E3 +data4 0x256FF4A2, 0x00000000 +data8 0x4002032CD37BBE04 +data4 0x25208AED, 0x00000000 +data8 0x4001A3BD7F050775 +data4 0x24B72DD6, 0x00000000 +data8 0x40014789A554848A +data4 0x24AB4DAA, 0x00000000 +data8 0x4000EE65323E81B7 +data4 0x2584C440, 0x00000000 +data8 0x4000982721CF1293 +data4 0x25C9428D, 0x00000000 +data8 0x400044A93D415EEB +data4 0x25DC8482, 0x00000000 +data8 0x3FFFE78FBD72C577 +data4 0x257F5070, 0x00000000 +data8 0x3FFF4AC375EFD28E +data4 0x23EBBF7A, 0x00000000 +data8 0x3FFEB2AF60B52DDE +data4 0x22EECA07, 0x00000000 +data8 0x3FFE1F1935204180 +data4 0x24191079, 0x00000000 +data8 0x3FFD8FCA54F7E60A +data4 0x248D3058, 0x00000000 +LOCAL_OBJECT_END(tanl_table_cm2) + +LOCAL_OBJECT_START(tanl_table_cm1) +// +// Entries C_hi double-precision memory format +// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) +// Entries C_lo single-precision memory format +// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) +// +data8 0x3FFCC06A79F6FADE +data4 0x239C7886, 0x00000000 +data8 0x3FFBB91F891662A6 +data4 0x250BD191, 0x00000000 +data8 0x3FFABFB6529F155D +data4 0x256CC3E6, 0x00000000 +data8 0x3FF9D3002E964AE9 +data4 0x250843E3, 0x00000000 +data8 0x3FF8F1EF89DCB383 +data4 0x2277C87E, 0x00000000 +data8 0x3FF81B937C87DBD6 +data4 0x256DA6CF, 0x00000000 +data8 0x3FF74F141042EDE4 +data4 0x2573D28A, 0x00000000 +data8 0x3FF68BAF1784B360 +data4 0x242E489A, 0x00000000 +data8 0x3FF5D0B57C923C4C +data4 0x2532D940, 0x00000000 +data8 0x3FF51D88F418EF20 +data4 0x253C7DD6, 0x00000000 +data8 0x3FF4719A02F88DAE +data4 0x23DB59BF, 0x00000000 +data8 0x3FF3CC6649DA0788 +data4 0x252B4756, 0x00000000 +data8 0x3FF32D770B980DB8 +data4 0x23FE585F, 0x00000000 +data8 0x3FF2945FE56C987A +data4 0x25378A63, 0x00000000 +data8 0x3FF200BDB16523F6 +data4 0x247BB2E0, 0x00000000 +data8 0x3FF172358CE27778 +data4 0x24446538, 0x00000000 +data8 0x3FF0E873FDEFE692 +data4 0x2514638F, 0x00000000 +data8 0x3FF0632C33154062 +data4 0x24A7FC27, 0x00000000 +data8 0x3FEFC42EB3EF115F +data4 0x248FD0FE, 0x00000000 +data8 0x3FEEC9E8135D26F6 +data4 0x2385C719, 0x00000000 +LOCAL_OBJECT_END(tanl_table_cm1) + +LOCAL_OBJECT_START(tanl_table_scim2) +// +// Entries SC_inv in Swapped IEEE format (extended) +// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64) +// +data8 0x839D6D4A1BF30C9E, 0x00004001 +data8 0x80092804554B0EB0, 0x00004001 +data8 0xF959F94CA1CF0DE9, 0x00004000 +data8 0xF3086BA077378677, 0x00004000 +data8 0xED154515CCD4723C, 0x00004000 +data8 0xE77909441C27CF25, 0x00004000 +data8 0xE22D037D8DDACB88, 0x00004000 +data8 0xDD2B2D8A89C73522, 0x00004000 +data8 0xD86E1A23BB2C1171, 0x00004000 +data8 0xD3F0E288DFF5E0F9, 0x00004000 +data8 0xCFAF16B1283BEBD5, 0x00004000 +data8 0xCBA4AFAA0D88DD53, 0x00004000 +data8 0xC7CE03CCCA67C43D, 0x00004000 +data8 0xC427BC820CA0DDB0, 0x00004000 +data8 0xC0AECD57F13D8CAB, 0x00004000 +data8 0xBD606C3871ECE6B1, 0x00004000 +data8 0xBA3A0A96A44C4929, 0x00004000 +data8 0xB7394F6FE5CCCEC1, 0x00004000 +data8 0xB45C12039637D8BC, 0x00004000 +data8 0xB1A0552892CB051B, 0x00004000 +data8 0xAF04432B6BA2FFD0, 0x00004000 +data8 0xAC862A237221235F, 0x00004000 +data8 0xAA2478AF5F00A9D1, 0x00004000 +data8 0xA7DDBB0C81E082BF, 0x00004000 +data8 0xA5B0987D45684FEE, 0x00004000 +data8 0xA39BD0F5627A8F53, 0x00004000 +data8 0xA19E3B036EC5C8B0, 0x00004000 +data8 0x9FB6C1F091CD7C66, 0x00004000 +data8 0x9DE464101FA3DF8A, 0x00004000 +data8 0x9C263139A8F6B888, 0x00004000 +data8 0x9A7B4968C27B0450, 0x00004000 +data8 0x98E2DB7E5EE614EE, 0x00004000 +LOCAL_OBJECT_END(tanl_table_scim2) + +LOCAL_OBJECT_START(tanl_table_scim1) +// +// Entries SC_inv in Swapped IEEE format (extended) +// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64) +// +data8 0x969F335C13B2B5BA, 0x00004000 +data8 0x93D446D9D4C0F548, 0x00004000 +data8 0x9147094F61B798AF, 0x00004000 +data8 0x8EF317CC758787AC, 0x00004000 +data8 0x8CD498B3B99EEFDB, 0x00004000 +data8 0x8AE82A7DDFF8BC37, 0x00004000 +data8 0x892AD546E3C55D42, 0x00004000 +data8 0x8799FEA9D15573C1, 0x00004000 +data8 0x86335F88435A4B4C, 0x00004000 +data8 0x84F4FB6E3E93A87B, 0x00004000 +data8 0x83DD195280A382FB, 0x00004000 +data8 0x82EA3D7FA4CB8C9E, 0x00004000 +data8 0x821B247C6861D0A8, 0x00004000 +data8 0x816EBED163E8D244, 0x00004000 +data8 0x80E42D9127E4CFC6, 0x00004000 +data8 0x807ABF8D28E64AFD, 0x00004000 +data8 0x8031EF26863B4FD8, 0x00004000 +data8 0x800960ADAE8C11FD, 0x00004000 +data8 0x8000E1475FDBEC21, 0x00004000 +data8 0x80186650A07791FA, 0x00004000 +LOCAL_OBJECT_END(tanl_table_scim1) + +Arg = f8 +Save_Norm_Arg = f8 // For input to reduction routine +Result = f8 +r = f8 // For output from reduction routine +c = f9 // For output from reduction routine +U_2 = f10 +rsq = f11 +C_hi = f12 +C_lo = f13 +T_hi = f14 +T_lo = f15 + +d_1 = f33 +N_0 = f34 +tail = f35 +tanx = f36 +Cx = f37 +Sx = f38 +sgn_r = f39 +CORR = f40 +P = f41 +D = f42 +ArgPrime = f43 +P_0 = f44 + +P2_1 = f45 +P2_2 = f46 +P2_3 = f47 + +P1_1 = f45 +P1_2 = f46 +P1_3 = f47 + +P1_4 = f48 +P1_5 = f49 +P1_6 = f50 +P1_7 = f51 +P1_8 = f52 +P1_9 = f53 + +x = f56 +xsq = f57 +Tx = f58 +Tx1 = f59 +Set = f60 +poly1 = f61 +poly2 = f62 +Poly = f63 +Poly1 = f64 +Poly2 = f65 +r_to_the_8 = f66 +B = f67 +SC_inv = f68 +Pos_r = f69 +N_0_fix = f70 +d_2 = f71 +PI_BY_4 = f72 +TWO_TO_NEG14 = f74 +TWO_TO_NEG33 = f75 +NEGTWO_TO_NEG14 = f76 +NEGTWO_TO_NEG33 = f77 +two_by_PI = f78 +N = f79 +N_fix = f80 +P_1 = f81 +P_2 = f82 +P_3 = f83 +s_val = f84 +w = f85 +B_mask1 = f86 +B_mask2 = f87 +w2 = f88 +A = f89 +a = f90 +t = f91 +U_1 = f92 +NEGTWO_TO_NEG2 = f93 +TWO_TO_NEG2 = f94 +Q1_1 = f95 +Q1_2 = f96 +Q1_3 = f97 +Q1_4 = f98 +Q1_5 = f99 +Q1_6 = f100 +Q1_7 = f101 +Q1_8 = f102 +S_hi = f103 +S_lo = f104 +V_hi = f105 +V_lo = f106 +U_hi = f107 +U_lo = f108 +U_hiabs = f109 +V_hiabs = f110 +V = f111 +Inv_P_0 = f112 + +FR_inv_pi_2to63 = f113 +FR_rshf_2to64 = f114 +FR_2tom64 = f115 +FR_rshf = f116 +Norm_Arg = f117 +Abs_Arg = f118 +TWO_TO_NEG65 = f119 +fp_tmp = f120 +mOne = f121 + +GR_SAVE_B0 = r33 +GR_SAVE_GP = r34 +GR_SAVE_PFS = r35 +table_base = r36 +table_ptr1 = r37 +table_ptr2 = r38 +table_ptr3 = r39 +lookup = r40 +N_fix_gr = r41 +GR_exp_2tom2 = r42 +GR_exp_2tom65 = r43 +exp_r = r44 +sig_r = r45 +bmask1 = r46 +table_offset = r47 +bmask2 = r48 +gr_tmp = r49 +cot_flag = r50 + +GR_sig_inv_pi = r51 +GR_rshf_2to64 = r52 +GR_exp_2tom64 = r53 +GR_rshf = r54 +GR_exp_2_to_63 = r55 +GR_exp_2_to_24 = r56 +GR_signexp_x = r57 +GR_exp_x = r58 +GR_exp_mask = r59 +GR_exp_2tom14 = r60 +GR_exp_m2tom14 = r61 +GR_exp_2tom33 = r62 +GR_exp_m2tom33 = r63 + +GR_SAVE_B0 = r64 +GR_SAVE_PFS = r65 +GR_SAVE_GP = r66 + +GR_Parameter_X = r67 +GR_Parameter_Y = r68 +GR_Parameter_RESULT = r69 +GR_Parameter_Tag = r70 + + +.section .text +.global __libm_tanl# +.global __libm_cotl# + +.proc __libm_cotl# +__libm_cotl: +.endp __libm_cotl# +LOCAL_LIBM_ENTRY(cotl) + +{ .mlx + alloc r32 = ar.pfs, 0,35,4,0 + movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi +} +{ .mlx + mov GR_exp_mask = 0x1ffff // Exponent mask + movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64) +} +;; + +// Check for NatVals, Infs , NaNs, and Zeros +{ .mfi + getf.exp GR_signexp_x = Arg // Get sign and exponent of x + fclass.m p6,p0 = Arg, 0x1E7 // Test for natval, nan, inf, zero + mov cot_flag = 0x1 +} +{ .mfb + addl table_base = @ltoff(TANL_BASE_CONSTANTS), gp // Pointer to table ptr + fnorm.s1 Norm_Arg = Arg // Normalize x + br.cond.sptk COMMON_PATH +};; + +LOCAL_LIBM_END(cotl) + + +.proc __libm_tanl# +__libm_tanl: +.endp __libm_tanl# +GLOBAL_IEEE754_ENTRY(tanl) + +{ .mlx + alloc r32 = ar.pfs, 0,35,4,0 + movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi +} +{ .mlx + mov GR_exp_mask = 0x1ffff // Exponent mask + movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64) +} +;; + +// Check for NatVals, Infs , NaNs, and Zeros +{ .mfi + getf.exp GR_signexp_x = Arg // Get sign and exponent of x + fclass.m p6,p0 = Arg, 0x1E7 // Test for natval, nan, inf, zero + mov cot_flag = 0x0 +} +{ .mfi + addl table_base = @ltoff(TANL_BASE_CONSTANTS), gp // Pointer to table ptr + fnorm.s1 Norm_Arg = Arg // Normalize x + nop.i 0 +};; + +// Common path for both tanl and cotl +COMMON_PATH: +{ .mfi + setf.sig FR_inv_pi_2to63 = GR_sig_inv_pi // Form 1/pi * 2^63 + fclass.m p9, p0 = Arg, 0x0b // Test x denormal + mov GR_exp_2tom64 = 0xffff - 64 // Scaling constant to compute N +} +{ .mlx + setf.d FR_rshf_2to64 = GR_rshf_2to64 // Form const 1.1000 * 2^(63+64) + movl GR_rshf = 0x43e8000000000000 // Form const 1.1000 * 2^63 +} +;; + +// Check for everything - if false, then must be pseudo-zero or pseudo-nan. +// Branch out to deal with special values. +{ .mfi + addl gr_tmp = -1,r0 + fclass.nm p7,p0 = Arg, 0x1FF // Test x unsupported + mov GR_exp_2_to_63 = 0xffff + 63 // Exponent of 2^63 +} +{ .mfb + ld8 table_base = [table_base] // Get pointer to constant table + fms.s1 mOne = f0, f0, f1 +(p6) br.cond.spnt TANL_SPECIAL // Branch if x natval, nan, inf, zero +} +;; + +{ .mmb + setf.sig fp_tmp = gr_tmp // Make a constant so fmpy produces inexact + mov GR_exp_2_to_24 = 0xffff + 24 // Exponent of 2^24 +(p9) br.cond.spnt TANL_DENORMAL // Branch if x denormal +} +;; + +TANL_COMMON: +// Return to here if x denormal +// +// Do fcmp to generate Denormal exception +// - can't do FNORM (will generate Underflow when U is unmasked!) +// Branch out to deal with unsupporteds values. +{ .mfi + setf.exp FR_2tom64 = GR_exp_2tom64 // Form 2^-64 for scaling N_float + fcmp.eq.s0 p0, p6 = Arg, f1 // Dummy to flag denormals + add table_ptr1 = 0, table_base // Point to tanl_table_1 +} +{ .mib + setf.d FR_rshf = GR_rshf // Form right shift const 1.1000 * 2^63 + add table_ptr2 = 80, table_base // Point to tanl_table_2 +(p7) br.cond.spnt TANL_UNSUPPORTED // Branch if x unsupported type +} +;; + +{ .mfi + and GR_exp_x = GR_exp_mask, GR_signexp_x // Get exponent of x + fmpy.s1 Save_Norm_Arg = Norm_Arg, f1 // Save x if large arg reduction + dep.z bmask1 = 0x7c, 56, 8 // Form mask to get 5 msb of r + // bmask1 = 0x7c00000000000000 +} +;; + +// +// Decide about the paths to take: +// Set PR_6 if |Arg| >= 2**63 +// Set PR_9 if |Arg| < 2**24 - CASE 1 OR 2 +// OTHERWISE Set PR_8 - CASE 3 OR 4 +// +// Branch out if the magnitude of the input argument is >= 2^63 +// - do this branch before the next. +{ .mfi + ldfe two_by_PI = [table_ptr1],16 // Load 2/pi + nop.f 999 + dep.z bmask2 = 0x41, 57, 7 // Form mask to OR to produce B + // bmask2 = 0x8200000000000000 +} +{ .mib + ldfe PI_BY_4 = [table_ptr2],16 // Load pi/4 + cmp.ge p6,p0 = GR_exp_x, GR_exp_2_to_63 // Is |x| >= 2^63 +(p6) br.cond.spnt TANL_ARG_TOO_LARGE // Branch if |x| >= 2^63 +} +;; + +{ .mmi + ldfe P_0 = [table_ptr1],16 // Load P_0 + ldfe Inv_P_0 = [table_ptr2],16 // Load Inv_P_0 + nop.i 999 +} +;; + +{ .mfi + ldfe P_1 = [table_ptr1],16 // Load P_1 + fmerge.s Abs_Arg = f0, Norm_Arg // Get |x| + mov GR_exp_m2tom33 = 0x2ffff - 33 // Form signexp of -2^-33 +} +{ .mfi + ldfe d_1 = [table_ptr2],16 // Load d_1 for 2^24 <= |x| < 2^63 + nop.f 999 + mov GR_exp_2tom33 = 0xffff - 33 // Form signexp of 2^-33 +} +;; + +{ .mmi + ldfe P_2 = [table_ptr1],16 // Load P_2 + ldfe d_2 = [table_ptr2],16 // Load d_2 for 2^24 <= |x| < 2^63 + cmp.ge p8,p0 = GR_exp_x, GR_exp_2_to_24 // Is |x| >= 2^24 +} +;; + +// Use special scaling to right shift so N=Arg * 2/pi is in rightmost bits +// Branch to Cases 3 or 4 if Arg <= -2**24 or Arg >= 2**24 +{ .mfb + ldfe P_3 = [table_ptr1],16 // Load P_3 + fma.s1 N_fix = Norm_Arg, FR_inv_pi_2to63, FR_rshf_2to64 +(p8) br.cond.spnt TANL_LARGER_ARG // Branch if 2^24 <= |x| < 2^63 +} +;; + +// Here if 0 < |x| < 2^24 +// ARGUMENT REDUCTION CODE - CASE 1 and 2 +// +{ .mmf + setf.exp TWO_TO_NEG33 = GR_exp_2tom33 // Form 2^-33 + setf.exp NEGTWO_TO_NEG33 = GR_exp_m2tom33 // Form -2^-33 + fmerge.s r = Norm_Arg,Norm_Arg // Assume r=x, ok if |x| < pi/4 +} +;; + +// +// If |Arg| < pi/4, set PR_8, else pi/4 <=|Arg| < 2^24 - set PR_9. +// +// Case 2: Convert integer N_fix back to normalized floating-point value. +{ .mfi + getf.sig sig_r = Norm_Arg // Get sig_r if 1/4 <= |x| < pi/4 + fcmp.lt.s1 p8,p9= Abs_Arg,PI_BY_4 // Test |x| < pi/4 + mov GR_exp_2tom2 = 0xffff - 2 // Form signexp of 2^-2 +} +{ .mfi + ldfps TWO_TO_NEG2, NEGTWO_TO_NEG2 = [table_ptr2] // Load 2^-2, -2^-2 + fms.s1 N = N_fix, FR_2tom64, FR_rshf // Use scaling to get N floated + mov N_fix_gr = r0 // Assume N=0, ok if |x| < pi/4 +} +;; + +// +// Case 1: Is |r| < 2**(-2). +// Arg is the same as r in this case. +// r = Arg +// c = 0 +// +// Case 2: Place integer part of N in GP register. +{ .mfi +(p9) getf.sig N_fix_gr = N_fix + fmerge.s c = f0, f0 // Assume c=0, ok if |x| < pi/4 + cmp.lt p10, p0 = GR_exp_x, GR_exp_2tom2 // Test if |x| < 1/4 +} +;; + +{ .mfi + setf.sig B_mask1 = bmask1 // Form mask to get 5 msb of r + nop.f 999 + mov exp_r = GR_exp_x // Get exp_r if 1/4 <= |x| < pi/4 +} +{ .mbb + setf.sig B_mask2 = bmask2 // Form mask to form B from r +(p10) br.cond.spnt TANL_SMALL_R // Branch if 0 < |x| < 1/4 +(p8) br.cond.spnt TANL_NORMAL_R // Branch if 1/4 <= |x| < pi/4 +} +;; + +// Here if pi/4 <= |x| < 2^24 +// +// Case 1: PR_3 is only affected when PR_1 is set. +// +// +// Case 2: w = N * P_2 +// Case 2: s_val = -N * P_1 + Arg +// + +{ .mfi + nop.m 999 + fnma.s1 s_val = N, P_1, Norm_Arg + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s1 w = N, P_2 // w = N * P_2 for |s| >= 2^-33 + nop.i 999 +} +;; + +// Case 2_reduce: w = N * P_3 (change sign) +{ .mfi + nop.m 999 + fmpy.s1 w2 = N, P_3 // w = N * P_3 for |s| < 2^-33 + nop.i 999 +} +;; + +// Case 1_reduce: r = s + w (change sign) +{ .mfi + nop.m 999 + fsub.s1 r = s_val, w // r = s_val - w for |s| >= 2^-33 + nop.i 999 +} +;; + +// Case 2_reduce: U_1 = N * P_2 + w +{ .mfi + nop.m 999 + fma.s1 U_1 = N, P_2, w2 // U_1 = N * P_2 + w for |s| < 2^-33 + nop.i 999 +} +;; + +// +// Decide between case_1 and case_2 reduce: +// Case 1_reduce: |s| >= 2**(-33) +// Case 2_reduce: |s| < 2**(-33) +// +{ .mfi + nop.m 999 + fcmp.lt.s1 p9, p8 = s_val, TWO_TO_NEG33 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 +(p9) fcmp.gt.s1 p9, p8 = s_val, NEGTWO_TO_NEG33 + nop.i 999 +} +;; + +// Case 1_reduce: c = s - r +{ .mfi + nop.m 999 + fsub.s1 c = s_val, r // c = s_val - r for |s| >= 2^-33 + nop.i 999 +} +;; + +// Case 2_reduce: r is complete here - continue to calculate c . +// r = s - U_1 +{ .mfi + nop.m 999 +(p9) fsub.s1 r = s_val, U_1 + nop.i 999 +} +{ .mfi + nop.m 999 +(p9) fms.s1 U_2 = N, P_2, U_1 + nop.i 999 +} +;; + +// +// Case 1_reduce: Is |r| < 2**(-2), if so set PR_10 +// else set PR_13. +// + +{ .mfi + nop.m 999 + fand B = B_mask1, r + nop.i 999 +} +{ .mfi + nop.m 999 +(p8) fcmp.lt.unc.s1 p10, p13 = r, TWO_TO_NEG2 + nop.i 999 +} +;; + +{ .mfi +(p8) getf.sig sig_r = r // Get signif of r if |s| >= 2^-33 + nop.f 999 + nop.i 999 +} +;; + +{ .mfi +(p8) getf.exp exp_r = r // Extract signexp of r if |s| >= 2^-33 +(p10) fcmp.gt.s1 p10, p13 = r, NEGTWO_TO_NEG2 + nop.i 999 +} +;; + +// Case 1_reduce: c is complete here. +// Case 1: Branch to SMALL_R or NORMAL_R. +// c = c + w (w has not been negated.) +{ .mfi + nop.m 999 +(p8) fsub.s1 c = c, w // c = c - w for |s| >= 2^-33 + nop.i 999 +} +{ .mbb + nop.m 999 +(p10) br.cond.spnt TANL_SMALL_R // Branch if pi/4 < |x| < 2^24 and |r|<1/4 +(p13) br.cond.sptk TANL_NORMAL_R_A // Branch if pi/4 < |x| < 2^24 and |r|>=1/4 +} +;; + + +// Here if pi/4 < |x| < 2^24 and |s| < 2^-33 +// +// Is i_1 = lsb of N_fix_gr even or odd? +// if i_1 == 0, set p11, else set p12. +// +{ .mfi + nop.m 999 + fsub.s1 s_val = s_val, r + add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl) +} +{ .mfi + nop.m 999 +// +// Case 2_reduce: +// U_2 = N * P_2 - U_1 +// Not needed until later. +// + fadd.s1 U_2 = U_2, w2 +// +// Case 2_reduce: +// s = s - r +// U_2 = U_2 + w +// + nop.i 999 +} +;; + +// +// Case 2_reduce: +// c = c - U_2 +// c is complete here +// Argument reduction ends here. +// +{ .mfi + nop.m 999 + fmpy.s1 rsq = r, r + tbit.z p11, p12 = N_fix_gr, 0 ;; // Set p11 if N even, p12 if odd +} + +{ .mfi + nop.m 999 +(p12) frcpa.s1 S_hi,p0 = f1, r + nop.i 999 +} +{ .mfi + nop.m 999 + fsub.s1 c = s_val, U_1 + nop.i 999 +} +;; + +{ .mmi + add table_ptr1 = 160, table_base ;; // Point to tanl_table_p1 + ldfe P1_1 = [table_ptr1],144 + nop.i 999 ;; +} +// +// Load P1_1 and point to Q1_1 . +// +{ .mfi + ldfe Q1_1 = [table_ptr1] +// +// N even: rsq = r * Z +// N odd: S_hi = frcpa(r) +// +(p12) fmerge.ns S_hi = S_hi, S_hi + nop.i 999 +} +{ .mfi + nop.m 999 +// +// Case 2_reduce: +// c = s - U_1 +// +(p9) fsub.s1 c = c, U_2 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: Change sign of S_hi +// +(p11) fmpy.s1 rsq = rsq, P1_1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: rsq = rsq * P1_1 +// N odd: poly1 = 1.0 + S_hi * r 16 bits partial account for necessary +// +(p11) fma.s1 Poly = r, rsq, c + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Poly = c + r * rsq +// N odd: S_hi = S_hi + S_hi*poly1 16 bits account for necessary +// +(p12) fma.s1 poly1 = S_hi, r, f1 +(p11) tbit.z.unc p14, p15 = cot_flag, 0 ;; // p14=1 for tanl; p15=1 for cotl +} +{ .mfi + nop.m 999 +// +// N even: Result = Poly + r +// N odd: poly1 = 1.0 + S_hi * r 32 bits partial +// +(p14) fadd.s0 Result = r, Poly // for tanl + nop.i 999 +} +{ .mfi + nop.m 999 +(p15) fms.s0 Result = r, mOne, Poly // for cotl + nop.i 999 +} +;; + +{ .mfi + nop.m 999 +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Result1 = Result + r +// N odd: S_hi = S_hi * poly1 + S_hi 32 bits +// +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * r + 1.0 64 bits partial +// +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * poly + 1.0 64 bits +// +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * r + 1.0 +// +(p12) fma.s1 poly1 = S_hi, c, poly1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * c + poly1 +// +(p12) fmpy.s1 S_lo = S_hi, poly1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: S_lo = S_hi * poly1 +// +(p12) fma.s1 S_lo = Q1_1, r, S_lo +(p12) tbit.z.unc p14, p15 = cot_flag, 0 // p14=1 for tanl; p15=1 for cotl +} +{ .mfi + nop.m 999 +// +// N odd: Result = S_hi + S_lo +// + fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: S_lo = S_lo + Q1_1 * r +// +(p14) fadd.s0 Result = S_hi, S_lo // for tanl + nop.i 999 +} +{ .mfb + nop.m 999 +(p15) fms.s0 Result = S_hi, mOne, S_lo // for cotl + br.ret.sptk b0 ;; // Exit for pi/4 <= |x| < 2^24 and |s| < 2^-33 +} + + +TANL_LARGER_ARG: +// Here if 2^24 <= |x| < 2^63 +// +// ARGUMENT REDUCTION CODE - CASE 3 and 4 +// + +{ .mmf + mov GR_exp_2tom14 = 0xffff - 14 // Form signexp of 2^-14 + mov GR_exp_m2tom14 = 0x2ffff - 14 // Form signexp of -2^-14 + fmpy.s1 N_0 = Norm_Arg, Inv_P_0 +} +;; + +{ .mmi + setf.exp TWO_TO_NEG14 = GR_exp_2tom14 // Form 2^-14 + setf.exp NEGTWO_TO_NEG14 = GR_exp_m2tom14// Form -2^-14 + nop.i 999 +} +;; + + +// +// Adjust table_ptr1 to beginning of table. +// N_0 = Arg * Inv_P_0 +// +{ .mmi + add table_ptr2 = 144, table_base ;; // Point to 2^-2 + ldfps TWO_TO_NEG2, NEGTWO_TO_NEG2 = [table_ptr2] + nop.i 999 +} +;; + +// +// N_0_fix = integer part of N_0 . +// +// +// Make N_0 the integer part. +// +{ .mfi + nop.m 999 + fcvt.fx.s1 N_0_fix = N_0 + nop.i 999 ;; +} +{ .mfi + setf.sig B_mask1 = bmask1 // Form mask to get 5 msb of r + fcvt.xf N_0 = N_0_fix + nop.i 999 ;; +} +{ .mfi + setf.sig B_mask2 = bmask2 // Form mask to form B from r + fnma.s1 ArgPrime = N_0, P_0, Norm_Arg + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s1 w = N_0, d_1 + nop.i 999 ;; +} +// +// ArgPrime = -N_0 * P_0 + Arg +// w = N_0 * d_1 +// +// +// N = ArgPrime * 2/pi +// +// fcvt.fx.s1 N_fix = N +// Use special scaling to right shift so N=Arg * 2/pi is in rightmost bits +// Branch to Cases 3 or 4 if Arg <= -2**24 or Arg >= 2**24 +{ .mfi + nop.m 999 + fma.s1 N_fix = ArgPrime, FR_inv_pi_2to63, FR_rshf_2to64 + + nop.i 999 ;; +} +// Convert integer N_fix back to normalized floating-point value. +{ .mfi + nop.m 999 + fms.s1 N = N_fix, FR_2tom64, FR_rshf // Use scaling to get N floated + nop.i 999 +} +;; + +// +// N is the integer part of the reduced-reduced argument. +// Put the integer in a GP register. +// +{ .mfi + getf.sig N_fix_gr = N_fix + nop.f 999 + nop.i 999 +} +;; + +// +// s_val = -N*P_1 + ArgPrime +// w = -N*P_2 + w +// +{ .mfi + nop.m 999 + fnma.s1 s_val = N, P_1, ArgPrime + nop.i 999 +} +{ .mfi + nop.m 999 + fnma.s1 w = N, P_2, w + nop.i 999 +} +;; + +// Case 4: V_hi = N * P_2 +// Case 4: U_hi = N_0 * d_1 +{ .mfi + nop.m 999 + fmpy.s1 V_hi = N, P_2 // V_hi = N * P_2 for |s| < 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s1 U_hi = N_0, d_1 // U_hi = N_0 * d_1 for |s| < 2^-14 + nop.i 999 +} +;; + +// Case 3: r = s_val + w (Z complete) +// Case 4: w = N * P_3 +{ .mfi + nop.m 999 + fadd.s1 r = s_val, w // r = s_val + w for |s| >= 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s1 w2 = N, P_3 // w = N * P_3 for |s| < 2^-14 + nop.i 999 +} +;; + +// Case 4: A = U_hi + V_hi +// Note: Worry about switched sign of V_hi, so subtract instead of add. +// Case 4: V_lo = -N * P_2 - V_hi (U_hi is in place of V_hi in writeup) +// Note: the (-) is still missing for V_hi. +{ .mfi + nop.m 999 + fsub.s1 A = U_hi, V_hi // A = U_hi - V_hi for |s| < 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 + fnma.s1 V_lo = N, P_2, V_hi // V_lo = V_hi - N * P_2 for |s| < 2^-14 + nop.i 999 +} +;; + +// Decide between case 3 and 4: +// Case 3: |s| >= 2**(-14) Set p10 +// Case 4: |s| < 2**(-14) Set p11 +// +// Case 4: U_lo = N_0 * d_1 - U_hi +{ .mfi + nop.m 999 + fms.s1 U_lo = N_0, d_1, U_hi // U_lo = N_0*d_1 - U_hi for |s| < 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 + fcmp.lt.s1 p11, p10 = s_val, TWO_TO_NEG14 + nop.i 999 +} +;; + +// Case 4: We need abs of both U_hi and V_hi - dont +// worry about switched sign of V_hi. +{ .mfi + nop.m 999 + fabs V_hiabs = V_hi // |V_hi| for |s| < 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 +(p11) fcmp.gt.s1 p11, p10 = s_val, NEGTWO_TO_NEG14 + nop.i 999 +} +;; + +// Case 3: c = s_val - r +{ .mfi + nop.m 999 + fabs U_hiabs = U_hi // |U_hi| for |s| < 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 + fsub.s1 c = s_val, r // c = s_val - r for |s| >= 2^-14 + nop.i 999 +} +;; + +// For Case 3, |s| >= 2^-14, determine if |r| < 1/4 +// +// Case 4: C_hi = s_val + A +// +{ .mfi + nop.m 999 +(p11) fadd.s1 C_hi = s_val, A // C_hi = s_val + A for |s| < 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 +(p10) fcmp.lt.unc.s1 p14, p15 = r, TWO_TO_NEG2 + nop.i 999 +} +;; + +{ .mfi + getf.sig sig_r = r // Get signif of r if |s| >= 2^-33 + fand B = B_mask1, r + nop.i 999 +} +;; + +// Case 4: t = U_lo + V_lo +{ .mfi + getf.exp exp_r = r // Extract signexp of r if |s| >= 2^-33 +(p11) fadd.s1 t = U_lo, V_lo // t = U_lo + V_lo for |s| < 2^-14 + nop.i 999 +} +{ .mfi + nop.m 999 +(p14) fcmp.gt.s1 p14, p15 = r, NEGTWO_TO_NEG2 + nop.i 999 +} +;; + +// Case 3: c = (s - r) + w (c complete) +{ .mfi + nop.m 999 +(p10) fadd.s1 c = c, w // c = c + w for |s| >= 2^-14 + nop.i 999 +} +{ .mbb + nop.m 999 +(p14) br.cond.spnt TANL_SMALL_R // Branch if 2^24 <= |x| < 2^63 and |r|< 1/4 +(p15) br.cond.sptk TANL_NORMAL_R_A // Branch if 2^24 <= |x| < 2^63 and |r|>=1/4 +} +;; + + +// Here if 2^24 <= |x| < 2^63 and |s| < 2^-14 >>>>>>> Case 4. +// +// Case 4: Set P_12 if U_hiabs >= V_hiabs +// Case 4: w = w + N_0 * d_2 +// Note: the (-) is now incorporated in w . +{ .mfi + add table_ptr1 = 160, table_base // Point to tanl_table_p1 + fcmp.ge.unc.s1 p12, p13 = U_hiabs, V_hiabs + nop.i 999 +} +{ .mfi + nop.m 999 + fms.s1 w2 = N_0, d_2, w2 + nop.i 999 +} +;; + +// Case 4: C_lo = s_val - C_hi +{ .mfi + ldfe P1_1 = [table_ptr1], 16 // Load P1_1 + fsub.s1 C_lo = s_val, C_hi + nop.i 999 +} +;; + +// +// Case 4: a = U_hi - A +// a = V_hi - A (do an add to account for missing (-) on V_hi +// +{ .mfi + ldfe P1_2 = [table_ptr1], 128 // Load P1_2 +(p12) fsub.s1 a = U_hi, A + nop.i 999 +} +{ .mfi + nop.m 999 +(p13) fadd.s1 a = V_hi, A + nop.i 999 +} +;; + +// Case 4: t = U_lo + V_lo + w +{ .mfi + ldfe Q1_1 = [table_ptr1], 16 // Load Q1_1 + fadd.s1 t = t, w2 + nop.i 999 +} +;; + +// Case 4: a = (U_hi - A) + V_hi +// a = (V_hi - A) + U_hi +// In each case account for negative missing form V_hi . +// +{ .mfi + ldfe Q1_2 = [table_ptr1], 16 // Load Q1_2 +(p12) fsub.s1 a = a, V_hi + nop.i 999 +} +{ .mfi + nop.m 999 +(p13) fsub.s1 a = U_hi, a + nop.i 999 +} +;; + +// +// Case 4: C_lo = (s_val - C_hi) + A +// +{ .mfi + nop.m 999 + fadd.s1 C_lo = C_lo, A + nop.i 999 ;; +} +// +// Case 4: t = t + a +// +{ .mfi + nop.m 999 + fadd.s1 t = t, a + nop.i 999 +} +;; + +// Case 4: C_lo = C_lo + t +// Case 4: r = C_hi + C_lo +{ .mfi + nop.m 999 + fadd.s1 C_lo = C_lo, t + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fadd.s1 r = C_hi, C_lo + nop.i 999 +} +;; + +// +// Case 4: c = C_hi - r +// +{ .mfi + nop.m 999 + fsub.s1 c = C_hi, r + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s1 rsq = r, r + add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl) +} +;; + +// Case 4: c = c + C_lo finished. +// +// Is i_1 = lsb of N_fix_gr even or odd? +// if i_1 == 0, set PR_11, else set PR_12. +// +{ .mfi + nop.m 999 + fadd.s1 c = c , C_lo + tbit.z p11, p12 = N_fix_gr, 0 +} +;; + +// r and c have been computed. +{ .mfi + nop.m 999 +(p12) frcpa.s1 S_hi, p0 = f1, r + nop.i 999 +} +{ .mfi + nop.m 999 +// +// N odd: Change sign of S_hi +// +(p11) fma.s1 Poly = rsq, P1_2, P1_1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 P = rsq, Q1_2, Q1_1 + nop.i 999 +} +{ .mfi + nop.m 999 +// +// N odd: Result = S_hi + S_lo (User supplied rounding mode for C1) +// + fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: rsq = r * r +// N odd: S_hi = frcpa(r) +// +(p12) fmerge.ns S_hi = S_hi, S_hi + nop.i 999 +} +{ .mfi + nop.m 999 +// +// N even: rsq = rsq * P1_2 + P1_1 +// N odd: poly1 = 1.0 + S_hi * r 16 bits partial account for necessary +// +(p11) fmpy.s1 Poly = rsq, Poly + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly1 = S_hi, r,f1 +(p11) tbit.z.unc p14, p15 = cot_flag, 0 // p14=1 for tanl; p15=1 for cotl +} +{ .mfi + nop.m 999 +// +// N even: Poly = Poly * rsq +// N odd: S_hi = S_hi + S_hi*poly1 16 bits account for necessary +// +(p11) fma.s1 Poly = r, Poly, c + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 +} +{ .mfi + nop.m 999 +// +// N odd: S_hi = S_hi * poly1 + S_hi 32 bits +// +(p14) fadd.s0 Result = r, Poly // for tanl + nop.i 999 ;; +} + +.pred.rel "mutex",p15,p12 +{ .mfi + nop.m 999 +(p15) fms.s0 Result = r, mOne, Poly // for cotl + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Poly = Poly * r + c +// N odd: poly1 = 1.0 + S_hi * r 32 bits partial +// +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Result = Poly + r (Rounding mode S0) +// N odd: poly1 = S_hi * r + 1.0 64 bits partial +// +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * poly + S_hi 64 bits +// +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * r + 1.0 +// +(p12) fma.s1 poly1 = S_hi, c, poly1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * c + poly1 +// +(p12) fmpy.s1 S_lo = S_hi, poly1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: S_lo = S_hi * poly1 +// +(p12) fma.s1 S_lo = P, r, S_lo +(p12) tbit.z.unc p14, p15 = cot_flag, 0 ;; // p14=1 for tanl; p15=1 for cotl +} + +{ .mfi + nop.m 999 +(p14) fadd.s0 Result = S_hi, S_lo // for tanl + nop.i 999 +} +{ .mfb + nop.m 999 +// +// N odd: S_lo = S_lo + r * P +// +(p15) fms.s0 Result = S_hi, mOne, S_lo // for cotl + br.ret.sptk b0 ;; // Exit for 2^24 <= |x| < 2^63 and |s| < 2^-14 +} + + +TANL_SMALL_R: +// Here if |r| < 1/4 +// r and c have been computed. +// ***************************************************************** +// ***************************************************************** +// ***************************************************************** +// N odd: S_hi = frcpa(r) +// Get [i_1] - lsb of N_fix_gr. Set p11 if N even, p12 if N odd. +// N even: rsq = r * r +{ .mfi + add table_ptr1 = 160, table_base // Point to tanl_table_p1 + frcpa.s1 S_hi, p0 = f1, r // S_hi for N odd + add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl) +} +{ .mfi + add table_ptr2 = 400, table_base // Point to Q1_7 + fmpy.s1 rsq = r, r + nop.i 999 +} +;; + +{ .mmi + ldfe P1_1 = [table_ptr1], 16 +;; + ldfe P1_2 = [table_ptr1], 16 + tbit.z p11, p12 = N_fix_gr, 0 +} +;; + + +{ .mfi + ldfe P1_3 = [table_ptr1], 96 + nop.f 999 + nop.i 999 +} +;; + +{ .mfi +(p11) ldfe P1_9 = [table_ptr1], -16 +(p12) fmerge.ns S_hi = S_hi, S_hi + nop.i 999 +} +{ .mfi + nop.m 999 +(p11) fmpy.s1 r_to_the_8 = rsq, rsq + nop.i 999 +} +;; + +// +// N even: Poly2 = P1_7 + Poly2 * rsq +// N odd: poly2 = Q1_5 + poly2 * rsq +// +{ .mfi +(p11) ldfe P1_8 = [table_ptr1], -16 +(p11) fadd.s1 CORR = rsq, f1 + nop.i 999 +} +;; + +// +// N even: Poly1 = P1_2 + P1_3 * rsq +// N odd: poly1 = 1.0 + S_hi * r +// 16 bits partial account for necessary (-1) +// +{ .mmi +(p11) ldfe P1_7 = [table_ptr1], -16 +;; +(p11) ldfe P1_6 = [table_ptr1], -16 + nop.i 999 +} +;; + +// +// N even: Poly1 = P1_1 + Poly1 * rsq +// N odd: S_hi = S_hi + S_hi * poly1) 16 bits account for necessary +// +// +// N even: Poly2 = P1_5 + Poly2 * rsq +// N odd: poly2 = Q1_3 + poly2 * rsq +// +{ .mfi +(p11) ldfe P1_5 = [table_ptr1], -16 +(p11) fmpy.s1 r_to_the_8 = r_to_the_8, r_to_the_8 + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 +} +;; + +// +// N even: Poly1 = Poly1 * rsq +// N odd: poly1 = 1.0 + S_hi * r 32 bits partial +// + +// +// N even: CORR = CORR * c +// N odd: S_hi = S_hi * poly1 + S_hi 32 bits +// + +// +// N even: Poly2 = P1_6 + Poly2 * rsq +// N odd: poly2 = Q1_4 + poly2 * rsq +// + +{ .mmf +(p11) ldfe P1_4 = [table_ptr1], -16 + nop.m 999 +(p11) fmpy.s1 CORR = CORR, c +} +;; + +{ .mfi + nop.m 999 +(p11) fma.s1 Poly1 = P1_3, rsq, P1_2 + nop.i 999 ;; +} +{ .mfi +(p12) ldfe Q1_7 = [table_ptr2], -16 +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 ;; +} +{ .mfi +(p12) ldfe Q1_6 = [table_ptr2], -16 +(p11) fma.s1 Poly2 = P1_9, rsq, P1_8 + nop.i 999 ;; +} +{ .mmi +(p12) ldfe Q1_5 = [table_ptr2], -16 ;; +(p12) ldfe Q1_4 = [table_ptr2], -16 + nop.i 999 ;; +} +{ .mfi +(p12) ldfe Q1_3 = [table_ptr2], -16 +// +// N even: Poly2 = P1_8 + P1_9 * rsq +// N odd: poly2 = Q1_6 + Q1_7 * rsq +// +(p11) fma.s1 Poly1 = Poly1, rsq, P1_1 + nop.i 999 ;; +} +{ .mfi +(p12) ldfe Q1_2 = [table_ptr2], -16 +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 ;; +} +{ .mfi +(p12) ldfe Q1_1 = [table_ptr2], -16 +(p11) fma.s1 Poly2 = Poly2, rsq, P1_7 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: CORR = rsq + 1 +// N even: r_to_the_8 = rsq * rsq +// +(p11) fmpy.s1 Poly1 = Poly1, rsq + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly2 = Q1_7, rsq, Q1_6 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p11) fma.s1 Poly2 = Poly2, rsq, P1_6 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly2 = poly2, rsq, Q1_5 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p11) fma.s1 Poly2= Poly2, rsq, P1_5 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 S_hi = S_hi, poly1, S_hi + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly2 = poly2, rsq, Q1_4 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: r_to_the_8 = r_to_the_8 * r_to_the_8 +// N odd: poly1 = S_hi * r + 1.0 64 bits partial +// +(p11) fma.s1 Poly2 = Poly2, rsq, P1_4 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Poly = CORR + Poly * r +// N odd: P = Q1_1 + poly2 * rsq +// +(p12) fma.s1 poly1 = S_hi, r, f1 + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly2 = poly2, rsq, Q1_3 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Poly2 = P1_4 + Poly2 * rsq +// N odd: poly2 = Q1_2 + poly2 * rsq +// +(p11) fma.s1 Poly = Poly2, r_to_the_8, Poly1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly1 = S_hi, c, poly1 + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fma.s1 poly2 = poly2, rsq, Q1_2 + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// N even: Poly = Poly1 + Poly2 * r_to_the_8 +// N odd: S_hi = S_hi * poly1 + S_hi 64 bits +// +(p11) fma.s1 Poly = Poly, r, CORR + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Result = r + Poly (User supplied rounding mode) +// N odd: poly1 = S_hi * c + poly1 +// +(p12) fmpy.s1 S_lo = S_hi, poly1 +(p11) tbit.z.unc p14, p15 = cot_flag, 0 // p14=1 for tanl; p15=1 for cotl +} +{ .mfi + nop.m 999 +(p12) fma.s1 P = poly2, rsq, Q1_1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: poly1 = S_hi * r + 1.0 +// +// +// N odd: S_lo = S_hi * poly1 +// +(p14) fadd.s0 Result = Poly, r // for tanl + nop.i 999 +} +{ .mfi + nop.m 999 +(p15) fms.s0 Result = Poly, mOne, r // for cotl + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +// +// N odd: S_lo = Q1_1 * c + S_lo +// +(p12) fma.s1 S_lo = Q1_1, c, S_lo + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: Result = S_lo + r * P +// +(p12) fma.s1 Result = P, r, S_lo +(p12) tbit.z.unc p14, p15 = cot_flag, 0 ;; // p14=1 for tanl; p15=1 for cotl +} + +// +// N odd: Result = Result + S_hi (user supplied rounding mode) +// +{ .mfi + nop.m 999 +(p14) fadd.s0 Result = Result, S_hi // for tanl + nop.i 999 +} +{ .mfb + nop.m 999 +(p15) fms.s0 Result = Result, mOne, S_hi // for cotl + br.ret.sptk b0 ;; // Exit |r| < 1/4 path +} + + +TANL_NORMAL_R: +// Here if 1/4 <= |x| < pi/4 or if |x| >= 2^63 and |r| >= 1/4 +// ******************************************************************* +// ******************************************************************* +// ******************************************************************* +// +// r and c have been computed. +// +{ .mfi + nop.m 999 + fand B = B_mask1, r + nop.i 999 +} +;; + +TANL_NORMAL_R_A: +// Enter here if pi/4 <= |x| < 2^63 and |r| >= 1/4 +// Get the 5 bits or r for the lookup. 1.xxxxx .... +{ .mmi + add table_ptr1 = 416, table_base // Point to tanl_table_p2 + mov GR_exp_2tom65 = 0xffff - 65 // Scaling constant for B + extr.u lookup = sig_r, 58, 5 +} +;; + +{ .mmi + ldfe P2_1 = [table_ptr1], 16 + setf.exp TWO_TO_NEG65 = GR_exp_2tom65 // 2^-65 for scaling B if exp_r=-2 + add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl) +} +;; + +.pred.rel "mutex",p11,p12 +// B = 2^63 * 1.xxxxx 100...0 +{ .mfi + ldfe P2_2 = [table_ptr1], 16 + for B = B_mask2, B + mov table_offset = 512 // Assume table offset is 512 +} +;; + +{ .mfi + ldfe P2_3 = [table_ptr1], 16 + fmerge.s Pos_r = f1, r + tbit.nz p8,p9 = exp_r, 0 +} +;; + +// Is B = 2** -2 or B= 2** -1? If 2**-1, then +// we want an offset of 512 for table addressing. +{ .mii + add table_ptr2 = 1296, table_base // Point to tanl_table_cm2 +(p9) shladd table_offset = lookup, 4, table_offset +(p8) shladd table_offset = lookup, 4, r0 +} +;; + +{ .mmi + add table_ptr1 = table_ptr1, table_offset // Point to T_hi + add table_ptr2 = table_ptr2, table_offset // Point to C_hi + add table_ptr3 = 2128, table_base // Point to tanl_table_scim2 +} +;; + +{ .mmi + ldfd T_hi = [table_ptr1], 8 // Load T_hi +;; + ldfd C_hi = [table_ptr2], 8 // Load C_hi + add table_ptr3 = table_ptr3, table_offset // Point to SC_inv +} +;; + +// +// x = |r| - B +// +// Convert B so it has the same exponent as Pos_r before subtracting +{ .mfi + ldfs T_lo = [table_ptr1] // Load T_lo +(p9) fnma.s1 x = B, FR_2tom64, Pos_r + nop.i 999 +} +{ .mfi + nop.m 999 +(p8) fnma.s1 x = B, TWO_TO_NEG65, Pos_r + nop.i 999 +} +;; + +{ .mfi + ldfs C_lo = [table_ptr2] // Load C_lo + nop.f 999 + nop.i 999 +} +;; + +{ .mfi + ldfe SC_inv = [table_ptr3] // Load SC_inv + fmerge.s sgn_r = r, f1 + tbit.z p11, p12 = N_fix_gr, 0 // p11 if N even, p12 if odd + +} +;; + +// +// xsq = x * x +// N even: Tx = T_hi * x +// +// N even: Tx1 = Tx + 1 +// N odd: Cx1 = 1 - Cx +// + +{ .mfi + nop.m 999 + fmpy.s1 xsq = x, x + nop.i 999 +} +{ .mfi + nop.m 999 +(p11) fmpy.s1 Tx = T_hi, x + nop.i 999 +} +;; + +// +// N odd: Cx = C_hi * x +// +{ .mfi + nop.m 999 +(p12) fmpy.s1 Cx = C_hi, x + nop.i 999 +} +;; +// +// N even and odd: P = P2_3 + P2_2 * xsq +// +{ .mfi + nop.m 999 + fma.s1 P = P2_3, xsq, P2_2 + nop.i 999 +} +{ .mfi + nop.m 999 +(p11) fadd.s1 Tx1 = Tx, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: D = C_hi - tanx +// N odd: D = T_hi + tanx +// +(p11) fmpy.s1 CORR = SC_inv, T_hi + nop.i 999 +} +{ .mfi + nop.m 999 + fmpy.s1 Sx = SC_inv, x + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fmpy.s1 CORR = SC_inv, C_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fsub.s1 V_hi = f1, Cx + nop.i 999 ;; +} +{ .mfi + nop.m 999 + fma.s1 P = P, xsq, P2_1 + nop.i 999 +} +{ .mfi + nop.m 999 +// +// N even and odd: P = P2_1 + P * xsq +// +(p11) fma.s1 V_hi = Tx, Tx1, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: Result = sgn_r * tail + T_hi (user rounding mode for C1) +// N odd: Result = sgn_r * tail + C_hi (user rounding mode for C1) +// + fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact + nop.i 999 ;; +} +{ .mfi + nop.m 999 + fmpy.s1 CORR = CORR, c + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fnma.s1 V_hi = Cx,V_hi,f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: V_hi = Tx * Tx1 + 1 +// N odd: Cx1 = 1 - Cx * Cx1 +// + fmpy.s1 P = P, xsq + nop.i 999 +} +{ .mfi + nop.m 999 +// +// N even and odd: P = P * xsq +// +(p11) fmpy.s1 V_hi = V_hi, T_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even and odd: tail = P * tail + V_lo +// +(p11) fmpy.s1 T_hi = sgn_r, T_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 + fmpy.s1 CORR = CORR, sgn_r + nop.i 999 ;; +} +{ .mfi + nop.m 999 +(p12) fmpy.s1 V_hi = V_hi,C_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: V_hi = T_hi * V_hi +// N odd: V_hi = C_hi * V_hi +// + fma.s1 tanx = P, x, x + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fnmpy.s1 C_hi = sgn_r, C_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: V_lo = 1 - V_hi + C_hi +// N odd: V_lo = 1 - V_hi + T_hi +// +(p11) fadd.s1 CORR = CORR, T_lo + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fsub.s1 CORR = CORR, C_lo + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even and odd: tanx = x + x * P +// N even and odd: Sx = SC_inv * x +// +(p11) fsub.s1 D = C_hi, tanx + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fadd.s1 D = T_hi, tanx + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N odd: CORR = SC_inv * C_hi +// N even: CORR = SC_inv * T_hi +// + fnma.s1 D = V_hi, D, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even and odd: D = 1 - V_hi * D +// N even and odd: CORR = CORR * c +// + fma.s1 V_hi = V_hi, D, V_hi + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even and odd: V_hi = V_hi + V_hi * D +// N even and odd: CORR = sgn_r * CORR +// +(p11) fnma.s1 V_lo = V_hi, C_hi, f1 + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fnma.s1 V_lo = V_hi, T_hi, f1 + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: CORR = COOR + T_lo +// N odd: CORR = CORR - C_lo +// +(p11) fma.s1 V_lo = tanx, V_hi, V_lo + tbit.nz p15, p0 = cot_flag, 0 // p15=1 if we compute cotl +} +{ .mfi + nop.m 999 +(p12) fnma.s1 V_lo = tanx, V_hi, V_lo + nop.i 999 ;; +} + +{ .mfi + nop.m 999 +(p15) fms.s1 T_hi = f0, f0, T_hi // to correct result's sign for cotl + nop.i 999 +} +{ .mfi + nop.m 999 +(p15) fms.s1 C_hi = f0, f0, C_hi // to correct result's sign for cotl + nop.i 999 +};; + +{ .mfi + nop.m 999 +(p15) fms.s1 sgn_r = f0, f0, sgn_r // to correct result's sign for cotl + nop.i 999 +};; + +{ .mfi + nop.m 999 +// +// N even: V_lo = V_lo + V_hi * tanx +// N odd: V_lo = V_lo - V_hi * tanx +// +(p11) fnma.s1 V_lo = C_lo, V_hi, V_lo + nop.i 999 +} +{ .mfi + nop.m 999 +(p12) fnma.s1 V_lo = T_lo, V_hi, V_lo + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: V_lo = V_lo - V_hi * C_lo +// N odd: V_lo = V_lo - V_hi * T_lo +// + fmpy.s1 V_lo = V_hi, V_lo + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even and odd: V_lo = V_lo * V_hi +// + fadd.s1 tail = V_hi, V_lo + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even and odd: tail = V_hi + V_lo +// + fma.s1 tail = tail, P, V_lo + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even: T_hi = sgn_r * T_hi +// N odd : C_hi = -sgn_r * C_hi +// + fma.s1 tail = tail, Sx, CORR + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even and odd: tail = Sx * tail + CORR +// + fma.s1 tail = V_hi, Sx, tail + nop.i 999 ;; +} +{ .mfi + nop.m 999 +// +// N even an odd: tail = Sx * V_hi + tail +// +(p11) fma.s0 Result = sgn_r, tail, T_hi + nop.i 999 +} +{ .mfb + nop.m 999 +(p12) fma.s0 Result = sgn_r, tail, C_hi + br.ret.sptk b0 ;; // Exit for 1/4 <= |r| < pi/4 +} + +TANL_DENORMAL: +// Here if x denormal +{ .mfb + getf.exp GR_signexp_x = Norm_Arg // Get sign and exponent of x + nop.f 999 + br.cond.sptk TANL_COMMON // Return to common code +} +;; + + +TANL_SPECIAL: +TANL_UNSUPPORTED: +// +// Code for NaNs, Unsupporteds, Infs, or +/- zero ? +// Invalid raised for Infs and SNaNs. +// + +{ .mfi + nop.m 999 + fmerge.s f10 = f8, f8 // Save input for error call + tbit.nz p6, p7 = cot_flag, 0 // p6=1 if we compute cotl +} +;; + +{ .mfi + nop.m 999 +(p6) fclass.m p6, p7 = f8, 0x7 // Test for zero (cotl only) + nop.i 999 +} +;; + +.pred.rel "mutex", p6, p7 +{ .mfi +(p6) mov GR_Parameter_Tag = 225 // (cotl) +(p6) frcpa.s0 f8, p0 = f1, f8 // cotl(+-0) = +-Inf + nop.i 999 +} +{ .mfb + nop.m 999 +(p7) fmpy.s0 f8 = f8, f0 +(p7) br.ret.sptk b0 +} +;; + +GLOBAL_IEEE754_END(tanl) + + +LOCAL_LIBM_ENTRY(__libm_error_region) +.prologue + +// (1) +{ .mfi + add GR_Parameter_Y=-32,sp // Parameter 2 value + nop.f 0 +.save ar.pfs,GR_SAVE_PFS + mov GR_SAVE_PFS=ar.pfs // Save ar.pfs +} +{ .mfi +.fframe 64 + add sp=-64,sp // Create new stack + nop.f 0 + mov GR_SAVE_GP=gp // Save gp +};; + +// (2) +{ .mmi + stfe [GR_Parameter_Y] = f1,16 // STORE Parameter 2 on stack + add GR_Parameter_X = 16,sp // Parameter 1 address +.save b0, GR_SAVE_B0 + mov GR_SAVE_B0=b0 // Save b0 +};; + +.body +// (3) +{ .mib + stfe [GR_Parameter_X] = f10 // STORE Parameter 1 on stack + add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address + nop.b 0 +} +{ .mib + stfe [GR_Parameter_Y] = f8 // STORE Parameter 3 on stack + add GR_Parameter_Y = -16,GR_Parameter_Y + br.call.sptk b0=__libm_error_support# // Call error handling function +};; +{ .mmi + nop.m 0 + nop.m 0 + add GR_Parameter_RESULT = 48,sp +};; + +// (4) +{ .mmi + ldfe f8 = [GR_Parameter_RESULT] // Get return result off stack +.restore sp + add sp = 64,sp // Restore stack pointer + mov b0 = GR_SAVE_B0 // Restore return address +};; +{ .mib + mov gp = GR_SAVE_GP // Restore gp + mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs + br.ret.sptk b0 // Return +};; + +LOCAL_LIBM_END(__libm_error_region) + +.type __libm_error_support#,@function +.global __libm_error_support# + + +// ******************************************************************* +// ******************************************************************* +// ******************************************************************* +// +// Special Code to handle very large argument case. +// Call int __libm_pi_by_2_reduce(x,r,c) for |arguments| >= 2**63 +// The interface is custom: +// On input: +// (Arg or x) is in f8 +// On output: +// r is in f8 +// c is in f9 +// N is in r8 +// We know also that __libm_pi_by_2_reduce preserves f10-15, f71-127. We +// use this to eliminate save/restore of key fp registers in this calling +// function. +// +// ******************************************************************* +// ******************************************************************* +// ******************************************************************* + +LOCAL_LIBM_ENTRY(__libm_callout) +TANL_ARG_TOO_LARGE: +.prologue +{ .mfi + add table_ptr2 = 144, table_base // Point to 2^-2 + nop.f 999 +.save ar.pfs,GR_SAVE_PFS + mov GR_SAVE_PFS=ar.pfs // Save ar.pfs +} +;; + +// Load 2^-2, -2^-2 +{ .mmi + ldfps TWO_TO_NEG2, NEGTWO_TO_NEG2 = [table_ptr2] + setf.sig B_mask1 = bmask1 // Form mask to get 5 msb of r +.save b0, GR_SAVE_B0 + mov GR_SAVE_B0=b0 // Save b0 +};; + +.body +// +// Call argument reduction with x in f8 +// Returns with N in r8, r in f8, c in f9 +// Assumes f71-127 are preserved across the call +// +{ .mib + setf.sig B_mask2 = bmask2 // Form mask to form B from r + mov GR_SAVE_GP=gp // Save gp + br.call.sptk b0=__libm_pi_by_2_reduce# +} +;; + +// +// Is |r| < 2**(-2) +// +{ .mfi + getf.sig sig_r = r // Extract significand of r + fcmp.lt.s1 p6, p0 = r, TWO_TO_NEG2 + mov gp = GR_SAVE_GP // Restore gp +} +;; + +{ .mfi + getf.exp exp_r = r // Extract signexp of r + nop.f 999 + mov b0 = GR_SAVE_B0 // Restore return address +} +;; + +// +// Get N_fix_gr +// +{ .mfi + mov N_fix_gr = r8 +(p6) fcmp.gt.unc.s1 p6, p0 = r, NEGTWO_TO_NEG2 + mov ar.pfs = GR_SAVE_PFS // Restore pfs +} +;; + +{ .mbb + nop.m 999 +(p6) br.cond.spnt TANL_SMALL_R // Branch if |r| < 1/4 + br.cond.sptk TANL_NORMAL_R // Branch if 1/4 <= |r| < pi/4 +} +;; + +LOCAL_LIBM_END(__libm_callout) + +.type __libm_pi_by_2_reduce#,@function +.global __libm_pi_by_2_reduce# |