/* Function asinhf vectorized with SSE4. Copyright (C) 2021-2022 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 https://www.gnu.org/licenses/. */ /* * ALGORITHM DESCRIPTION: * * Compute asinh(x) as log(x + sqrt(x*x + 1)) * * Special cases: * * asinh(NaN) = quiet NaN, and raise invalid exception * asinh(INF) = that INF * asinh(0) = that 0 * */ /* Offsets for data table __svml_sasinh_data_internal */ #define SgnMask 0 #define sOne 16 #define sPoly 32 #define iBrkValue 160 #define iOffExpoMask 176 #define sBigThreshold 192 #define sC2 208 #define sC3 224 #define sHalf 240 #define sLargestFinite 256 #define sLittleThreshold 272 #define sSign 288 #define sThirtyOne 304 #define sTopMask11 320 #define sTopMask8 336 #define XScale 352 #define sLn2 368 #include .section .text.sse4, "ax", @progbits ENTRY(_ZGVbN4v_asinhf_sse4) subq $72, %rsp cfi_def_cfa_offset(80) movaps %xmm0, %xmm8 /* * Split X into high and low parts, XHi (<= 11 bits) and XLo (<= 13 bits) * We could use either X or |X| here, but it doesn't seem to matter */ movups sTopMask11+__svml_sasinh_data_internal(%rip), %xmm10 movaps %xmm8, %xmm2 andps %xmm8, %xmm10 /* * Compute X^2 = (XHi + XLo)^2 = XHi^2 + XLo * (X + XHi) * The two parts are shifted off by around 11 bits. So even though * the low bit will not in general be exact, it's near enough */ movaps %xmm10, %xmm3 subps %xmm10, %xmm2 mulps %xmm10, %xmm3 addps %xmm8, %xmm10 /* Load the constant 1 and a sign mask */ movups sOne+__svml_sasinh_data_internal(%rip), %xmm7 /* * Finally, express Y + W = X^2 + 1 accurately where Y has <= 8 bits. * If |X| <= 1 then |XHi| <= 1 and so |X2Hi| <= 1, so we can treat 1 * as the dominant component in the compensated summation. Otherwise, * if |X| >= 1, then since X2Hi only has 22 significant bits, the basic * addition will be exact anyway until we get to |X| >= 2^24. But by * that time the log function is well-conditioned enough that the * rounding error doesn't matter. Hence we can treat 1 as dominant even * if it literally isn't. */ movaps %xmm7, %xmm11 movaps %xmm7, %xmm4 movups sTopMask8+__svml_sasinh_data_internal(%rip), %xmm12 addps %xmm3, %xmm11 mulps %xmm10, %xmm2 subps %xmm11, %xmm4 movaps %xmm12, %xmm0 addps %xmm3, %xmm4 /* * Unfortunately, we can still be in trouble if |X| <= 2^-5, since * the absolute error 2^-(7+24)-ish in sqrt(1 + X^2) gets scaled up * by 1/X and comes close to our threshold. Hence if |X| <= 2^-4, * perform an alternative computation * sqrt(1 + X^2) - 1 = X^2/2 - X^4/8 + X^6/16 * X2 = X^2 */ addps %xmm2, %xmm3 addps %xmm2, %xmm4 andps %xmm11, %xmm0 /* * Compute R = 1/sqrt(Y + W) * (1 + d) * Force R to <= 8 significant bits. * This means that R * Y and R^2 * Y are exactly representable. */ rsqrtps %xmm0, %xmm14 subps %xmm0, %xmm11 andps %xmm12, %xmm14 addps %xmm11, %xmm4 /* * Compute S = (Y/sqrt(Y + W)) * (1 + d) * and T = (W/sqrt(Y + W)) * (1 + d) * so that S + T = sqrt(Y + W) * (1 + d) * S is exact, and the rounding error in T is OK. */ mulps %xmm14, %xmm0 mulps %xmm14, %xmm4 /* * Get the absolute value of the input, since we will exploit antisymmetry * and mostly assume X >= 0 in the core computation */ movups SgnMask+__svml_sasinh_data_internal(%rip), %xmm6 /* * Compute e = -(2 * d + d^2) * The first FMR is exact, and the rounding error in the other is acceptable * since d and e are ~ 2^-8 */ movaps %xmm14, %xmm13 andps %xmm8, %xmm6 /* * Obtain sqrt(1 + X^2) - 1 in two pieces * sqrt(1 + X^2) - 1 * = sqrt(Y + W) - 1 * = (S + T) * (1 + Corr) - 1 * = [S - 1] + [T + (S + T) * Corr] * We need a compensated summation for the last part. We treat S - 1 * as the larger part; it certainly is until about X < 2^-4, and in that * case, the error is affordable since X dominates over sqrt(1 + X^2) - 1 * Final sum is dTmp5 (hi) + dTmp7 (lo) */ movaps %xmm0, %xmm1 /* * Check whether the input is finite, by checking |X| <= MaxFloat * Otherwise set the rangemask so that the callout will get used. * Note that this will also use the callout for NaNs since not(NaN <= MaxFloat) */ movaps %xmm6, %xmm9 /* * The following computation can go wrong for very large X, basically * because X^2 overflows. But for large X we have * asinh(X) / log(2 X) - 1 =~= 1/(4 * X^2), so for X >= 2^30 * we can just later stick X back into the log and tweak up the exponent. * Actually we scale X by 2^-30 and tweak the exponent up by 31, * to stay in the safe range for the later log computation. * Compute a flag now telling us when do do this. */ movaps %xmm6, %xmm5 cmpnleps sLargestFinite+__svml_sasinh_data_internal(%rip), %xmm9 cmpltps sBigThreshold+__svml_sasinh_data_internal(%rip), %xmm5 mulps %xmm0, %xmm13 addps %xmm4, %xmm1 subps %xmm7, %xmm0 mulps %xmm4, %xmm14 movmskps %xmm9, %edx movaps %xmm7, %xmm9 /* * Now 1 / (1 + d) * = 1 / (1 + (sqrt(1 - e) - 1)) * = 1 / sqrt(1 - e) * = 1 + 1/2 * e + 3/8 * e^2 + 5/16 * e^3 + 35/128 * e^4 + ... * So compute the first three nonconstant terms of that, so that * we have a relative correction (1 + Corr) to apply to S etc. * C1 = 1/2 * C2 = 3/8 * C3 = 5/16 */ movups sC3+__svml_sasinh_data_internal(%rip), %xmm15 subps %xmm13, %xmm9 movups sHalf+__svml_sasinh_data_internal(%rip), %xmm10 subps %xmm14, %xmm9 /* sX2over2 = X^2/2 */ mulps %xmm10, %xmm3 mulps %xmm9, %xmm15 /* sX46 = -X^4/4 + X^6/8 */ movaps %xmm3, %xmm2 movaps %xmm3, %xmm12 /* * Now do another compensated sum to add |X| + [sqrt(1 + X^2) - 1]. * It's always safe to assume |X| is larger. * This is the final 2-part argument to the log1p function */ movaps %xmm6, %xmm14 addps sC2+__svml_sasinh_data_internal(%rip), %xmm15 mulps %xmm9, %xmm15 addps %xmm10, %xmm15 mulps %xmm15, %xmm9 mulps %xmm1, %xmm9 /* Now multiplex to the case X = 2^-30 * input, Xl = sL = 0 in the "big" case. */ movups XScale+__svml_sasinh_data_internal(%rip), %xmm15 addps %xmm9, %xmm4 movaps %xmm4, %xmm11 addps %xmm0, %xmm11 subps %xmm11, %xmm0 addps %xmm0, %xmm4 /* sX4over4 = X^4/4 */ movaps %xmm3, %xmm0 mulps %xmm3, %xmm0 mulps %xmm0, %xmm2 subps %xmm0, %xmm2 /* * Now we feed into the log1p code, using H in place of _VARG1 and * also adding L into Xl. * compute 1+x as high, low parts */ movaps %xmm7, %xmm0 /* sX46over2 = -X^4/8 + x^6/16 */ mulps %xmm2, %xmm10 movaps %xmm7, %xmm2 addps %xmm10, %xmm12 subps %xmm12, %xmm3 addps %xmm3, %xmm10 /* Now multiplex the two possible computations */ movaps %xmm6, %xmm3 cmpleps sLittleThreshold+__svml_sasinh_data_internal(%rip), %xmm3 movaps %xmm3, %xmm13 andps %xmm3, %xmm12 andnps %xmm11, %xmm13 movaps %xmm3, %xmm1 orps %xmm12, %xmm13 andnps %xmm4, %xmm1 andps %xmm3, %xmm10 movaps %xmm6, %xmm4 orps %xmm10, %xmm1 addps %xmm13, %xmm14 mulps %xmm15, %xmm6 maxps %xmm14, %xmm0 minps %xmm14, %xmm2 subps %xmm14, %xmm4 movaps %xmm0, %xmm3 addps %xmm4, %xmm13 addps %xmm2, %xmm3 addps %xmm13, %xmm1 subps %xmm3, %xmm0 movaps %xmm5, %xmm4 andps %xmm5, %xmm3 andnps %xmm6, %xmm4 addps %xmm0, %xmm2 /* * Now resume the main code. * reduction: compute r, n */ movdqu iBrkValue+__svml_sasinh_data_internal(%rip), %xmm6 orps %xmm3, %xmm4 psubd %xmm6, %xmm4 movaps %xmm7, %xmm0 addps %xmm2, %xmm1 movdqu iOffExpoMask+__svml_sasinh_data_internal(%rip), %xmm2 pand %xmm4, %xmm2 psrad $23, %xmm4 cvtdq2ps %xmm4, %xmm3 pslld $23, %xmm4 andps %xmm5, %xmm1 paddd %xmm6, %xmm2 psubd %xmm4, %xmm0 mulps %xmm0, %xmm1 /* polynomial evaluation */ subps %xmm7, %xmm2 movups sPoly+112+__svml_sasinh_data_internal(%rip), %xmm7 addps %xmm2, %xmm1 mulps %xmm1, %xmm7 movaps %xmm5, %xmm2 /* Add 31 to the exponent in the "large" case to get log(2 * input) */ movups sThirtyOne+__svml_sasinh_data_internal(%rip), %xmm0 addps sPoly+96+__svml_sasinh_data_internal(%rip), %xmm7 addps %xmm3, %xmm0 mulps %xmm1, %xmm7 andnps %xmm0, %xmm2 andps %xmm5, %xmm3 orps %xmm3, %xmm2 addps sPoly+80+__svml_sasinh_data_internal(%rip), %xmm7 /* final reconstruction */ mulps sLn2+__svml_sasinh_data_internal(%rip), %xmm2 mulps %xmm1, %xmm7 /* Finally, reincorporate the original sign. */ movups sSign+__svml_sasinh_data_internal(%rip), %xmm0 andps %xmm8, %xmm0 addps sPoly+64+__svml_sasinh_data_internal(%rip), %xmm7 mulps %xmm1, %xmm7 addps sPoly+48+__svml_sasinh_data_internal(%rip), %xmm7 mulps %xmm1, %xmm7 addps sPoly+32+__svml_sasinh_data_internal(%rip), %xmm7 mulps %xmm1, %xmm7 addps sPoly+16+__svml_sasinh_data_internal(%rip), %xmm7 mulps %xmm1, %xmm7 addps sPoly+__svml_sasinh_data_internal(%rip), %xmm7 mulps %xmm1, %xmm7 mulps %xmm1, %xmm7 addps %xmm7, %xmm1 addps %xmm2, %xmm1 pxor %xmm1, %xmm0 testl %edx, %edx /* Go to special inputs processing branch */ jne L(SPECIAL_VALUES_BRANCH) # LOE rbx rbp r12 r13 r14 r15 edx xmm0 xmm8 /* Restore registers * and exit the function */ L(EXIT): addq $72, %rsp cfi_def_cfa_offset(8) ret cfi_def_cfa_offset(80) /* Branch to process * special inputs */ L(SPECIAL_VALUES_BRANCH): movups %xmm8, 32(%rsp) movups %xmm0, 48(%rsp) # LOE rbx rbp r12 r13 r14 r15 edx xorl %eax, %eax movq %r12, 16(%rsp) cfi_offset(12, -64) movl %eax, %r12d movq %r13, 8(%rsp) cfi_offset(13, -72) movl %edx, %r13d movq %r14, (%rsp) cfi_offset(14, -80) # LOE rbx rbp r15 r12d r13d /* Range mask * bits check */ L(RANGEMASK_CHECK): btl %r12d, %r13d /* Call scalar math function */ jc L(SCALAR_MATH_CALL) # LOE rbx rbp r15 r12d r13d /* Special inputs * processing loop */ L(SPECIAL_VALUES_LOOP): incl %r12d cmpl $4, %r12d /* Check bits in range mask */ jl L(RANGEMASK_CHECK) # LOE rbx rbp r15 r12d r13d movq 16(%rsp), %r12 cfi_restore(12) movq 8(%rsp), %r13 cfi_restore(13) movq (%rsp), %r14 cfi_restore(14) movups 48(%rsp), %xmm0 /* Go to exit */ jmp L(EXIT) cfi_offset(12, -64) cfi_offset(13, -72) cfi_offset(14, -80) # LOE rbx rbp r12 r13 r14 r15 xmm0 /* Scalar math fucntion call * to process special input */ L(SCALAR_MATH_CALL): movl %r12d, %r14d movss 32(%rsp, %r14, 4), %xmm0 call asinhf@PLT # LOE rbx rbp r14 r15 r12d r13d xmm0 movss %xmm0, 48(%rsp, %r14, 4) /* Process special inputs in loop */ jmp L(SPECIAL_VALUES_LOOP) # LOE rbx rbp r15 r12d r13d END(_ZGVbN4v_asinhf_sse4) .section .rodata, "a" .align 16 #ifdef __svml_sasinh_data_internal_typedef typedef unsigned int VUINT32; typedef struct { __declspec(align(16)) VUINT32 SgnMask[4][1]; __declspec(align(16)) VUINT32 sOne[4][1]; __declspec(align(16)) VUINT32 sPoly[8][4][1]; __declspec(align(16)) VUINT32 iBrkValue[4][1]; __declspec(align(16)) VUINT32 iOffExpoMask[4][1]; __declspec(align(16)) VUINT32 sBigThreshold[4][1]; __declspec(align(16)) VUINT32 sC2[4][1]; __declspec(align(16)) VUINT32 sC3[4][1]; __declspec(align(16)) VUINT32 sHalf[4][1]; __declspec(align(16)) VUINT32 sLargestFinite[4][1]; __declspec(align(16)) VUINT32 sLittleThreshold[4][1]; __declspec(align(16)) VUINT32 sSign[4][1]; __declspec(align(16)) VUINT32 sThirtyOne[4][1]; __declspec(align(16)) VUINT32 sTopMask11[4][1]; __declspec(align(16)) VUINT32 sTopMask8[4][1]; __declspec(align(16)) VUINT32 XScale[4][1]; __declspec(align(16)) VUINT32 sLn2[4][1]; } __svml_sasinh_data_internal; #endif __svml_sasinh_data_internal: /* SgnMask */ .long 0x7fffffff, 0x7fffffff, 0x7fffffff, 0x7fffffff /* sOne = SP 1.0 */ .align 16 .long 0x3f800000, 0x3f800000, 0x3f800000, 0x3f800000 /* sPoly[] = SP polynomial */ .align 16 .long 0xbf000000, 0xbf000000, 0xbf000000, 0xbf000000 /* -5.0000000000000000000000000e-01 P0 */ .long 0x3eaaaa94, 0x3eaaaa94, 0x3eaaaa94, 0x3eaaaa94 /* 3.3333265781402587890625000e-01 P1 */ .long 0xbe80058e, 0xbe80058e, 0xbe80058e, 0xbe80058e /* -2.5004237890243530273437500e-01 P2 */ .long 0x3e4ce190, 0x3e4ce190, 0x3e4ce190, 0x3e4ce190 /* 2.0007920265197753906250000e-01 P3 */ .long 0xbe28ad37, 0xbe28ad37, 0xbe28ad37, 0xbe28ad37 /* -1.6472326219081878662109375e-01 P4 */ .long 0x3e0fcb12, 0x3e0fcb12, 0x3e0fcb12, 0x3e0fcb12 /* 1.4042308926582336425781250e-01 P5 */ .long 0xbe1ad9e3, 0xbe1ad9e3, 0xbe1ad9e3, 0xbe1ad9e3 /* -1.5122179687023162841796875e-01 P6 */ .long 0x3e0d84ed, 0x3e0d84ed, 0x3e0d84ed, 0x3e0d84ed /* 1.3820238411426544189453125e-01 P7 */ /* iBrkValue = SP 2/3 */ .align 16 .long 0x3f2aaaab, 0x3f2aaaab, 0x3f2aaaab, 0x3f2aaaab /* iOffExpoMask = SP significand mask */ .align 16 .long 0x007fffff, 0x007fffff, 0x007fffff, 0x007fffff /* sBigThreshold */ .align 16 .long 0x4E800000, 0x4E800000, 0x4E800000, 0x4E800000 /* sC2 */ .align 16 .long 0x3EC00000, 0x3EC00000, 0x3EC00000, 0x3EC00000 /* sC3 */ .align 16 .long 0x3EA00000, 0x3EA00000, 0x3EA00000, 0x3EA00000 /* sHalf */ .align 16 .long 0x3F000000, 0x3F000000, 0x3F000000, 0x3F000000 /* sLargestFinite */ .align 16 .long 0x7F7FFFFF, 0x7F7FFFFF, 0x7F7FFFFF, 0x7F7FFFFF /* sLittleThreshold */ .align 16 .long 0x3D800000, 0x3D800000, 0x3D800000, 0x3D800000 /* sSign */ .align 16 .long 0x80000000, 0x80000000, 0x80000000, 0x80000000 /* sThirtyOne */ .align 16 .long 0x41F80000, 0x41F80000, 0x41F80000, 0x41F80000 /* sTopMask11 */ .align 16 .long 0xFFFFE000, 0xFFFFE000, 0xFFFFE000, 0xFFFFE000 /* sTopMask8 */ .align 16 .long 0xFFFF0000, 0xFFFF0000, 0xFFFF0000, 0xFFFF0000 /* XScale */ .align 16 .long 0x30800000, 0x30800000, 0x30800000, 0x30800000 /* sLn2 = SP ln(2) */ .align 16 .long 0x3f317218, 0x3f317218, 0x3f317218, 0x3f317218 .align 16 .type __svml_sasinh_data_internal, @object .size __svml_sasinh_data_internal, .-__svml_sasinh_data_internal