/* Optimized cosf function. Copyright (C) 2012-2017 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 <http://www.gnu.org/licenses/>. */ #include <sysdep.h> #include <errno.h> /* Short algorithm description: * * 1) if |x| == 0: return 1.0-|x|. * 2) if |x| < 2^-27: return 1.0-|x|. * 3) if |x| < 2^-5 : return 1.0+x^2*DP_COS2_0+x^5*DP_COS2_1. * 4) if |x| < Pi/4: return 1.0+x^2*(C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4)))). * 5) if |x| < 9*Pi/4: * 5.1) Range reduction: k=trunc(|x|/(Pi/4)), j=(k+1)&0x0e, n=k+3, * t=|x|-j*Pi/4. * 5.2) Reconstruction: * s = (-1.0)^((n>>2)&1) * if(n&2 != 0) { * using cos(t) polynomial for |t|<Pi/4, result is * s * (1.0+t^2*(C0+t^2*(C1+t^2*(C2+t^2*(C3+t^2*C4))))). * } else { * using sin(t) polynomial for |t|<Pi/4, result is * s * t * (1.0+t^2*(S0+t^2*(S1+t^2*(S2+t^2*(S3+t^2*S4))))). * } * 6) if |x| < 2^23, large args: * 6.1) Range reduction: k=trunc(|x|/(Pi/4)), j=(k+1)&0xfffffffe, n=k+3, * t=|x|-j*Pi/4. * 6.2) Reconstruction same as (5.2). * 7) if |x| >= 2^23, very large args: * 7.1) Range reduction: k=trunc(|x|/(Pi/4)), j=(k+1)&0xfffffffe, n=k+3, * t=|x|-j*Pi/4. * 7.2) Reconstruction same as (5.2). * 8) if x is Inf, return x-x, and set errno=EDOM. * 9) if x is NaN, return x-x. * * Special cases: * cos(+-0) = 1 not raising inexact, * cos(subnormal) raises inexact, * cos(min_normalized) raises inexact, * cos(normalized) raises inexact, * cos(Inf) = NaN, raises invalid, sets errno to EDOM, * cos(NaN) = NaN. */ .text ENTRY(__cosf) /* Input: single precision x in %xmm0 */ movd %xmm0, %eax /* Bits of x */ movaps %xmm0, %xmm7 /* Copy of x */ cvtss2sd %xmm0, %xmm0 /* DP x */ movss L(SP_ABS_MASK)(%rip), %xmm3 andl $0x7fffffff, %eax /* |x| */ cmpl $0x3f490fdb, %eax /* |x|<Pi/4? */ jb L(arg_less_pio4) /* Here if |x|>=Pi/4 */ andps %xmm7, %xmm3 /* SP |x| */ andpd L(DP_ABS_MASK)(%rip), %xmm0 /* DP |x| */ movss L(SP_INVPIO4)(%rip), %xmm2 /* SP 1/(Pi/4) */ cmpl $0x40e231d6, %eax /* |x|<9*Pi/4? */ jae L(large_args) /* Here if Pi/4<=|x|<9*Pi/4 */ mulss %xmm3, %xmm2 /* SP |x|/(Pi/4) */ cvttss2si %xmm2, %eax /* k, number of Pi/4 in x */ lea L(PIO4J)(%rip), %rsi addl $1, %eax /* k+1 */ movl $0x0e, %edx andl %eax, %edx /* j = (k+1)&0x0e */ addl $2, %eax /* n */ subsd (%rsi,%rdx,8), %xmm0 /* t = |x| - j * Pi/4 */ L(reconstruction): /* Input: %eax=n, %xmm0=t */ testl $2, %eax /* n&2 != 0? */ jz L(sin_poly) /*L(cos_poly):*/ /* Here if cos(x) calculated using cos(t) polynomial for |t|<Pi/4: * y = t*t; z = y*y; * s = sign(x) * (-1.0)^((n>>2)&1) * result = s * (1.0+t^2*(C0+t^2*(C1+t^2*(C2+t^2*(C3+t^2*C4))))) */ shrl $2, %eax /* n>>2 */ mulsd %xmm0, %xmm0 /* y=t^2 */ andl $1, %eax /* (n>>2)&1 */ movaps %xmm0, %xmm1 /* y */ mulsd %xmm0, %xmm0 /* z=t^4 */ movsd L(DP_C4)(%rip), %xmm4 /* C4 */ mulsd %xmm0, %xmm4 /* z*C4 */ movsd L(DP_C3)(%rip), %xmm3 /* C3 */ mulsd %xmm0, %xmm3 /* z*C3 */ lea L(DP_ONES)(%rip), %rsi addsd L(DP_C2)(%rip), %xmm4 /* C2+z*C4 */ mulsd %xmm0, %xmm4 /* z*(C2+z*C4) */ addsd L(DP_C1)(%rip), %xmm3 /* C1+z*C3 */ mulsd %xmm0, %xmm3 /* z*(C1+z*C3) */ addsd L(DP_C0)(%rip), %xmm4 /* C0+z*(C2+z*C4) */ mulsd %xmm1, %xmm4 /* y*(C0+z*(C2+z*C4)) */ addsd %xmm4, %xmm3 /* y*(C0+y*(C1+y*(C2+y*(C3+y*C4)))) */ /* 1.0+y*(C0+y*(C1+y*(C2+y*(C3+y*C4)))) */ addsd L(DP_ONES)(%rip), %xmm3 mulsd (%rsi,%rax,8), %xmm3 /* DP result */ cvtsd2ss %xmm3, %xmm0 /* SP result */ ret .p2align 4 L(sin_poly): /* Here if cos(x) calculated using sin(t) polynomial for |t|<Pi/4: * y = t*t; z = y*y; * s = sign(x) * (-1.0)^((n>>2)&1) * result = s * t * (1.0+t^2*(S0+t^2*(S1+t^2*(S2+t^2*(S3+t^2*S4))))) */ movaps %xmm0, %xmm4 /* t */ shrl $2, %eax /* n>>2 */ mulsd %xmm0, %xmm0 /* y=t^2 */ andl $1, %eax /* (n>>2)&1 */ movaps %xmm0, %xmm1 /* y */ mulsd %xmm0, %xmm0 /* z=t^4 */ movsd L(DP_S4)(%rip), %xmm2 /* S4 */ mulsd %xmm0, %xmm2 /* z*S4 */ movsd L(DP_S3)(%rip), %xmm3 /* S3 */ mulsd %xmm0, %xmm3 /* z*S3 */ lea L(DP_ONES)(%rip), %rsi addsd L(DP_S2)(%rip), %xmm2 /* S2+z*S4 */ mulsd %xmm0, %xmm2 /* z*(S2+z*S4) */ addsd L(DP_S1)(%rip), %xmm3 /* S1+z*S3 */ mulsd %xmm0, %xmm3 /* z*(S1+z*S3) */ addsd L(DP_S0)(%rip), %xmm2 /* S0+z*(S2+z*S4) */ mulsd %xmm1, %xmm2 /* y*(S0+z*(S2+z*S4)) */ /* t*s, where s = sign(x) * (-1.0)^((n>>2)&1) */ mulsd (%rsi,%rax,8), %xmm4 /* y*(S0+y*(S1+y*(S2+y*(S3+y*S4)))) */ addsd %xmm2, %xmm3 /* t*s*y*(S0+y*(S1+y*(S2+y*(S3+y*S4)))) */ mulsd %xmm4, %xmm3 /* t*s*(1.0+y*(S0+y*(S1+y*(S2+y*(S3+y*S4)))) */ addsd %xmm4, %xmm3 cvtsd2ss %xmm3, %xmm0 /* SP result */ ret .p2align 4 L(large_args): /* Here if |x|>=9*Pi/4 */ cmpl $0x7f800000, %eax /* x is Inf or NaN? */ jae L(arg_inf_or_nan) /* Here if finite |x|>=9*Pi/4 */ cmpl $0x4b000000, %eax /* |x|<2^23? */ jae L(very_large_args) /* Here if 9*Pi/4<=|x|<2^23 */ movsd L(DP_INVPIO4)(%rip), %xmm1 /* 1/(Pi/4) */ mulsd %xmm0, %xmm1 /* |x|/(Pi/4) */ cvttsd2si %xmm1, %eax /* k=trunc(|x|/(Pi/4)) */ addl $1, %eax /* k+1 */ movl %eax, %edx andl $0xfffffffe, %edx /* j=(k+1)&0xfffffffe */ cvtsi2sdl %edx, %xmm4 /* DP j */ movsd L(DP_PIO4HI)(%rip), %xmm2 /* -PIO4HI = high part of -Pi/4 */ mulsd %xmm4, %xmm2 /* -j*PIO4HI */ movsd L(DP_PIO4LO)(%rip), %xmm3 /* -PIO4LO = low part of -Pi/4 */ addsd %xmm2, %xmm0 /* |x| - j*PIO4HI */ addl $2, %eax /* n */ mulsd %xmm3, %xmm4 /* j*PIO4LO */ addsd %xmm4, %xmm0 /* t = |x| - j*PIO4HI - j*PIO4LO */ jmp L(reconstruction) .p2align 4 L(very_large_args): /* Here if finite |x|>=2^23 */ /* bitpos = (ix>>23) - BIAS_32 + 59; */ shrl $23, %eax /* eb = biased exponent of x */ /* bitpos = eb - 0x7f + 59, where 0x7f is exponent bias */ subl $68, %eax movl $28, %ecx /* %cl=28 */ movl %eax, %edx /* bitpos copy */ /* j = bitpos/28; */ div %cl /* j in register %al=%ax/%cl */ movapd %xmm0, %xmm3 /* |x| */ /* clear unneeded remainder from %ah */ andl $0xff, %eax imull $28, %eax, %ecx /* j*28 */ lea L(_FPI)(%rip), %rsi movsd L(DP_HI_MASK)(%rip), %xmm4 /* DP_HI_MASK */ movapd %xmm0, %xmm5 /* |x| */ mulsd -16(%rsi,%rax,8), %xmm3 /* tmp3 = FPI[j-2]*|x| */ movapd %xmm0, %xmm1 /* |x| */ mulsd -8(%rsi,%rax,8), %xmm5 /* tmp2 = FPI[j-1]*|x| */ mulsd (%rsi,%rax,8), %xmm0 /* tmp0 = FPI[j]*|x| */ addl $19, %ecx /* j*28+19 */ mulsd 8(%rsi,%rax,8), %xmm1 /* tmp1 = FPI[j+1]*|x| */ cmpl %ecx, %edx /* bitpos>=j*28+19? */ jl L(very_large_skip1) /* Here if bitpos>=j*28+19 */ andpd %xmm3, %xmm4 /* HI(tmp3) */ subsd %xmm4, %xmm3 /* tmp3 = tmp3 - HI(tmp3) */ L(very_large_skip1): movsd L(DP_2POW52)(%rip), %xmm6 movapd %xmm5, %xmm2 /* tmp2 copy */ addsd %xmm3, %xmm5 /* tmp5 = tmp3 + tmp2 */ movl $1, %edx addsd %xmm5, %xmm6 /* tmp6 = tmp5 + 2^52 */ movsd 8+L(DP_2POW52)(%rip), %xmm4 movd %xmm6, %eax /* k = I64_LO(tmp6); */ addsd %xmm6, %xmm4 /* tmp4 = tmp6 - 2^52 */ comisd %xmm5, %xmm4 /* tmp4 > tmp5? */ jbe L(very_large_skip2) /* Here if tmp4 > tmp5 */ subl $1, %eax /* k-- */ addsd 8+L(DP_ONES)(%rip), %xmm4 /* tmp4 -= 1.0 */ L(very_large_skip2): andl %eax, %edx /* k&1 */ lea L(DP_ZERONE)(%rip), %rsi subsd %xmm4, %xmm3 /* tmp3 -= tmp4 */ addsd (%rsi,%rdx,8), %xmm3 /* t = DP_ZERONE[k&1] + tmp3 */ addsd %xmm2, %xmm3 /* t += tmp2 */ addsd %xmm3, %xmm0 /* t += tmp0 */ addl $3, %eax /* n=k+3 */ addsd %xmm1, %xmm0 /* t += tmp1 */ mulsd L(DP_PIO4)(%rip), %xmm0 /* t *= PI04 */ jmp L(reconstruction) /* end of very_large_args peth */ .p2align 4 L(arg_less_pio4): /* Here if |x|<Pi/4 */ cmpl $0x3d000000, %eax /* |x|<2^-5? */ jl L(arg_less_2pn5) /* Here if 2^-5<=|x|<Pi/4 */ mulsd %xmm0, %xmm0 /* y=x^2 */ movaps %xmm0, %xmm1 /* y */ mulsd %xmm0, %xmm0 /* z=x^4 */ movsd L(DP_C4)(%rip), %xmm3 /* C4 */ mulsd %xmm0, %xmm3 /* z*C4 */ movsd L(DP_C3)(%rip), %xmm5 /* C3 */ mulsd %xmm0, %xmm5 /* z*C3 */ addsd L(DP_C2)(%rip), %xmm3 /* C2+z*C4 */ mulsd %xmm0, %xmm3 /* z*(C2+z*C4) */ addsd L(DP_C1)(%rip), %xmm5 /* C1+z*C3 */ mulsd %xmm0, %xmm5 /* z*(C1+z*C3) */ addsd L(DP_C0)(%rip), %xmm3 /* C0+z*(C2+z*C4) */ mulsd %xmm1, %xmm3 /* y*(C0+z*(C2+z*C4)) */ /* y*(C0+y*(C1+y*(C2+y*(C3+y*C4)))) */ addsd %xmm5, %xmm3 /* 1.0 + y*(C0+y*(C1+y*(C2+y*(C3+y*C4)))) */ addsd L(DP_ONES)(%rip), %xmm3 cvtsd2ss %xmm3, %xmm0 /* SP result */ ret .p2align 4 L(arg_less_2pn5): /* Here if |x|<2^-5 */ cmpl $0x32000000, %eax /* |x|<2^-27? */ jl L(arg_less_2pn27) /* Here if 2^-27<=|x|<2^-5 */ mulsd %xmm0, %xmm0 /* DP x^2 */ movsd L(DP_COS2_1)(%rip), %xmm3 /* DP DP_COS2_1 */ mulsd %xmm0, %xmm3 /* DP x^2*DP_COS2_1 */ addsd L(DP_COS2_0)(%rip), %xmm3 /* DP DP_COS2_0+x^2*DP_COS2_1 */ mulsd %xmm0, %xmm3 /* DP x^2*DP_COS2_0+x^4*DP_COS2_1 */ /* DP 1.0+x^2*DP_COS2_0+x^4*DP_COS2_1 */ addsd L(DP_ONES)(%rip), %xmm3 cvtsd2ss %xmm3, %xmm0 /* SP result */ ret .p2align 4 L(arg_less_2pn27): /* Here if |x|<2^-27 */ andps L(SP_ABS_MASK)(%rip),%xmm7 /* |x| */ movss L(SP_ONE)(%rip), %xmm0 /* 1.0 */ subss %xmm7, %xmm0 /* result is 1.0-|x| */ ret .p2align 4 L(arg_inf_or_nan): /* Here if |x| is Inf or NAN */ jne L(skip_errno_setting) /* in case of x is NaN */ /* Align stack to 16 bytes. */ subq $8, %rsp cfi_adjust_cfa_offset (8) /* Here if x is Inf. Set errno to EDOM. */ call JUMPTARGET(__errno_location) addq $8, %rsp cfi_adjust_cfa_offset (-8) movl $EDOM, (%rax) .p2align 4 L(skip_errno_setting): /* Here if |x| is Inf or NAN. Continued. */ movaps %xmm7, %xmm0 /* load x */ subss %xmm0, %xmm0 /* Result is NaN */ ret END(__cosf) .section .rodata, "a" .p2align 3 L(PIO4J): /* Table of j*Pi/4, for j=0,1,..,10 */ .long 0x00000000,0x00000000 .long 0x54442d18,0x3fe921fb .long 0x54442d18,0x3ff921fb .long 0x7f3321d2,0x4002d97c .long 0x54442d18,0x400921fb .long 0x2955385e,0x400f6a7a .long 0x7f3321d2,0x4012d97c .long 0xe9bba775,0x4015fdbb .long 0x54442d18,0x401921fb .long 0xbeccb2bb,0x401c463a .long 0x2955385e,0x401f6a7a .type L(PIO4J), @object ASM_SIZE_DIRECTIVE(L(PIO4J)) .p2align 3 L(_FPI): /* 4/Pi broken into sum of positive DP values */ .long 0x00000000,0x00000000 .long 0x6c000000,0x3ff45f30 .long 0x2a000000,0x3e3c9c88 .long 0xa8000000,0x3c54fe13 .long 0xd0000000,0x3aaf47d4 .long 0x6c000000,0x38fbb81b .long 0xe0000000,0x3714acc9 .long 0x7c000000,0x3560e410 .long 0x56000000,0x33bca2c7 .long 0xac000000,0x31fbd778 .long 0xe0000000,0x300b7246 .long 0xe8000000,0x2e5d2126 .long 0x48000000,0x2c970032 .long 0xe8000000,0x2ad77504 .long 0xe0000000,0x290921cf .long 0xb0000000,0x274deb1c .long 0xe0000000,0x25829a73 .long 0xbe000000,0x23fd1046 .long 0x10000000,0x2224baed .long 0x8e000000,0x20709d33 .long 0x80000000,0x1e535a2f .long 0x64000000,0x1cef904e .long 0x30000000,0x1b0d6398 .long 0x24000000,0x1964ce7d .long 0x16000000,0x17b908bf .type L(_FPI), @object ASM_SIZE_DIRECTIVE(L(_FPI)) /* Coefficients of polynomial for cos(x)~=1.0+x^2*DP_COS2_0+x^4*DP_COS2_1, |x|<2^-5. */ .p2align 3 L(DP_COS2_0): .long 0xff5cc6fd,0xbfdfffff .type L(DP_COS2_0), @object ASM_SIZE_DIRECTIVE(L(DP_COS2_0)) .p2align 3 L(DP_COS2_1): .long 0xb178dac5,0x3fa55514 .type L(DP_COS2_1), @object ASM_SIZE_DIRECTIVE(L(DP_COS2_1)) .p2align 3 L(DP_ZERONE): .long 0x00000000,0x00000000 /* 0.0 */ .long 0x00000000,0xbff00000 /* 1.0 */ .type L(DP_ZERONE), @object ASM_SIZE_DIRECTIVE(L(DP_ZERONE)) .p2align 3 L(DP_ONES): .long 0x00000000,0x3ff00000 /* +1.0 */ .long 0x00000000,0xbff00000 /* -1.0 */ .type L(DP_ONES), @object ASM_SIZE_DIRECTIVE(L(DP_ONES)) /* Coefficients of polynomial for sin(t)~=t+t^3*(S0+t^2*(S1+t^2*(S2+t^2*(S3+t^2*S4)))), |t|<Pi/4. */ .p2align 3 L(DP_S3): .long 0x64e6b5b4,0x3ec71d72 .type L(DP_S3), @object ASM_SIZE_DIRECTIVE(L(DP_S3)) .p2align 3 L(DP_S1): .long 0x10c2688b,0x3f811111 .type L(DP_S1), @object ASM_SIZE_DIRECTIVE(L(DP_S1)) .p2align 3 L(DP_S4): .long 0x1674b58a,0xbe5a947e .type L(DP_S4), @object ASM_SIZE_DIRECTIVE(L(DP_S4)) .p2align 3 L(DP_S2): .long 0x8b4bd1f9,0xbf2a019f .type L(DP_S2),@object ASM_SIZE_DIRECTIVE(L(DP_S2)) .p2align 3 L(DP_S0): .long 0x55551cd9,0xbfc55555 .type L(DP_S0), @object ASM_SIZE_DIRECTIVE(L(DP_S0)) /* Coefficients of polynomial for cos(t)~=1.0+t^2*(C0+t^2*(C1+t^2*(C2+t^2*(C3+t^2*C4)))), |t|<Pi/4. */ .p2align 3 L(DP_C3): .long 0x9ac43cc0,0x3efa00eb .type L(DP_C3), @object ASM_SIZE_DIRECTIVE(L(DP_C3)) .p2align 3 L(DP_C1): .long 0x545c50c7,0x3fa55555 .type L(DP_C1), @object ASM_SIZE_DIRECTIVE(L(DP_C1)) .p2align 3 L(DP_C4): .long 0xdd8844d7,0xbe923c97 .type L(DP_C4), @object ASM_SIZE_DIRECTIVE(L(DP_C4)) .p2align 3 L(DP_C2): .long 0x348b6874,0xbf56c16b .type L(DP_C2), @object ASM_SIZE_DIRECTIVE(L(DP_C2)) .p2align 3 L(DP_C0): .long 0xfffe98ae,0xbfdfffff .type L(DP_C0), @object ASM_SIZE_DIRECTIVE(L(DP_C0)) .p2align 3 L(DP_PIO4): .long 0x54442d18,0x3fe921fb /* Pi/4 */ .type L(DP_PIO4), @object ASM_SIZE_DIRECTIVE(L(DP_PIO4)) .p2align 3 L(DP_2POW52): .long 0x00000000,0x43300000 /* +2^52 */ .long 0x00000000,0xc3300000 /* -2^52 */ .type L(DP_2POW52), @object ASM_SIZE_DIRECTIVE(L(DP_2POW52)) .p2align 3 L(DP_INVPIO4): .long 0x6dc9c883,0x3ff45f30 /* 4/Pi */ .type L(DP_INVPIO4), @object ASM_SIZE_DIRECTIVE(L(DP_INVPIO4)) .p2align 3 L(DP_PIO4HI): .long 0x54000000,0xbfe921fb /* High part of Pi/4 */ .type L(DP_PIO4HI), @object ASM_SIZE_DIRECTIVE(L(DP_PIO4HI)) .p2align 3 L(DP_PIO4LO): .long 0x11A62633,0xbe010b46 /* Low part of Pi/4 */ .type L(DP_PIO4LO), @object ASM_SIZE_DIRECTIVE(L(DP_PIO4LO)) .p2align 2 L(SP_INVPIO4): .long 0x3fa2f983 /* 4/Pi */ .type L(SP_INVPIO4), @object ASM_SIZE_DIRECTIVE(L(SP_INVPIO4)) .p2align 4 L(DP_ABS_MASK): /* Mask for getting DP absolute value */ .long 0xffffffff,0x7fffffff .long 0xffffffff,0x7fffffff .type L(DP_ABS_MASK), @object ASM_SIZE_DIRECTIVE(L(DP_ABS_MASK)) .p2align 3 L(DP_HI_MASK): /* Mask for getting high 21 bits of DP value */ .long 0x00000000,0xffffffff .type L(DP_HI_MASK), @object ASM_SIZE_DIRECTIVE(L(DP_HI_MASK)) .p2align 4 L(SP_ABS_MASK): /* Mask for getting SP absolute value */ .long 0x7fffffff,0x7fffffff .long 0x7fffffff,0x7fffffff .type L(SP_ABS_MASK), @object ASM_SIZE_DIRECTIVE(L(SP_ABS_MASK)) .p2align 2 L(SP_ONE): .long 0x3f800000 /* 1.0 */ .type L(SP_ONE), @object ASM_SIZE_DIRECTIVE(L(SP_ONE)) weak_alias(__cosf, cosf)