/* Single-precision floating point square root. Copyright (C) 1997-2015 Free Software Foundation, Inc. This file is part of the GNU C Library. The GNU C Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. The GNU C Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU C Library; if not, see . */ #include #include #include #include #include #include #include static const float almost_half = 0.50000006; /* 0.5 + 2^-24 */ static const ieee_float_shape_type a_nan = {.word = 0x7fc00000 }; static const ieee_float_shape_type a_inf = {.word = 0x7f800000 }; static const float two48 = 281474976710656.0; static const float twom24 = 5.9604644775390625e-8; extern const float __t_sqrt[1024]; /* The method is based on a description in Computation of elementary functions on the IBM RISC System/6000 processor, P. W. Markstein, IBM J. Res. Develop, 34(1) 1990. Basically, it consists of two interleaved Newton-Raphson approximations, one to find the actual square root, and one to find its reciprocal without the expense of a division operation. The tricky bit here is the use of the POWER/PowerPC multiply-add operation to get the required accuracy with high speed. The argument reduction works by a combination of table lookup to obtain the initial guesses, and some careful modification of the generated guesses (which mostly runs on the integer unit, while the Newton-Raphson is running on the FPU). */ float __slow_ieee754_sqrtf (float x) { const float inf = a_inf.value; if (x > 0) { if (x != inf) { /* Variables named starting with 's' exist in the argument-reduced space, so that 2 > sx >= 0.5, 1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... . Variables named ending with 'i' are integer versions of floating-point values. */ float sx; /* The value of which we're trying to find the square root. */ float sg, g; /* Guess of the square root of x. */ float sd, d; /* Difference between the square of the guess and x. */ float sy; /* Estimate of 1/2g (overestimated by 1ulp). */ float sy2; /* 2*sy */ float e; /* Difference between y*g and 1/2 (note that e==se). */ float shx; /* == sx * fsg */ float fsg; /* sg*fsg == g. */ fenv_t fe; /* Saved floating-point environment (stores rounding mode and whether the inexact exception is enabled). */ uint32_t xi, sxi, fsgi; const float *t_sqrt; GET_FLOAT_WORD (xi, x); fe = fegetenv_register (); relax_fenv_state (); sxi = (xi & 0x3fffffff) | 0x3f000000; SET_FLOAT_WORD (sx, sxi); t_sqrt = __t_sqrt + (xi >> (23 - 8 - 1) & 0x3fe); sg = t_sqrt[0]; sy = t_sqrt[1]; /* Here we have three Newton-Raphson iterations each of a division and a square root and the remainder of the argument reduction, all interleaved. */ sd = -(sg * sg - sx); fsgi = (xi + 0x40000000) >> 1 & 0x7f800000; sy2 = sy + sy; sg = sy * sd + sg; /* 16-bit approximation to sqrt(sx). */ e = -(sy * sg - almost_half); SET_FLOAT_WORD (fsg, fsgi); sd = -(sg * sg - sx); sy = sy + e * sy2; if ((xi & 0x7f800000) == 0) goto denorm; shx = sx * fsg; sg = sg + sy * sd; /* 32-bit approximation to sqrt(sx), but perhaps rounded incorrectly. */ sy2 = sy + sy; g = sg * fsg; e = -(sy * sg - almost_half); d = -(g * sg - shx); sy = sy + e * sy2; fesetenv_register (fe); return g + sy * d; denorm: /* For denormalised numbers, we normalise, calculate the square root, and return an adjusted result. */ fesetenv_register (fe); return __slow_ieee754_sqrtf (x * two48) * twom24; } } else if (x < 0) { /* For some reason, some PowerPC32 processors don't implement FE_INVALID_SQRT. */ #ifdef FE_INVALID_SQRT feraiseexcept (FE_INVALID_SQRT); fenv_union_t u = { .fenv = fegetenv_register () }; if ((u.l & FE_INVALID) == 0) #endif feraiseexcept (FE_INVALID); x = a_nan.value; } return f_washf (x); } #undef __ieee754_sqrtf float __ieee754_sqrtf (float x) { double z; /* If the CPU is 64-bit we can use the optional FP instructions. */ if (__CPU_HAS_FSQRT) { /* Volatile is required to prevent the compiler from moving the fsqrt instruction above the branch. */ __asm __volatile (" fsqrts %0,%1\n" :"=f" (z):"f" (x)); } else z = __slow_ieee754_sqrtf (x); return z; } strong_alias (__ieee754_sqrtf, __sqrtf_finite)