/* * Written by J.T. Conklin <jtc@netbsd.org>. * Public domain. * * Adapted for `long double' by Ulrich Drepper <drepper@cygnus.com>. */ /* * The 8087 method for the exponential function is to calculate * exp(x) = 2^(x log2(e)) * after separating integer and fractional parts * x log2(e) = i + f, |f| <= .5 * 2^i is immediate but f needs to be precise for long double accuracy. * Suppress range reduction error in computing f by the following. * Separate x into integer and fractional parts * x = xi + xf, |xf| <= .5 * Separate log2(e) into the sum of an exact number c0 and small part c1. * c0 + c1 = log2(e) to extra precision * Then * f = (c0 xi - i) + c0 xf + c1 x * where c0 xi is exact and so also is (c0 xi - i). * -- moshier@na-net.ornl.gov */ #include <libm-alias-ldouble.h> #include <machine/asm.h> #include <x86_64-math-asm.h> #include <libm-alias-finite.h> #ifdef USE_AS_EXP10L # define IEEE754_EXPL __ieee754_exp10l # define EXPL_FINITE __exp10l_finite # define FLDLOG fldl2t #elif defined USE_AS_EXPM1L # define IEEE754_EXPL __expm1l # undef EXPL_FINITE # define FLDLOG fldl2e #else # define IEEE754_EXPL __ieee754_expl # define EXPL_FINITE __expl_finite # define FLDLOG fldl2e #endif .section .rodata.cst16,"aM",@progbits,16 .p2align 4 #ifdef USE_AS_EXP10L .type c0,@object c0: .byte 0, 0, 0, 0, 0, 0, 0x9a, 0xd4, 0x00, 0x40 .byte 0, 0, 0, 0, 0, 0 ASM_SIZE_DIRECTIVE(c0) .type c1,@object c1: .byte 0x58, 0x92, 0xfc, 0x15, 0x37, 0x9a, 0x97, 0xf0, 0xef, 0x3f .byte 0, 0, 0, 0, 0, 0 ASM_SIZE_DIRECTIVE(c1) #else .type c0,@object c0: .byte 0, 0, 0, 0, 0, 0, 0xaa, 0xb8, 0xff, 0x3f .byte 0, 0, 0, 0, 0, 0 ASM_SIZE_DIRECTIVE(c0) .type c1,@object c1: .byte 0x20, 0xfa, 0xee, 0xc2, 0x5f, 0x70, 0xa5, 0xec, 0xed, 0x3f .byte 0, 0, 0, 0, 0, 0 ASM_SIZE_DIRECTIVE(c1) #endif #ifndef USE_AS_EXPM1L .type csat,@object csat: .byte 0, 0, 0, 0, 0, 0, 0, 0x80, 0x0e, 0x40 .byte 0, 0, 0, 0, 0, 0 ASM_SIZE_DIRECTIVE(csat) DEFINE_LDBL_MIN #endif #ifdef PIC # define MO(op) op##(%rip) #else # define MO(op) op #endif .text ENTRY(IEEE754_EXPL) #ifdef USE_AS_EXPM1L movzwl 8+8(%rsp), %eax xorb $0x80, %ah // invert sign bit (now 1 is "positive") cmpl $0xc006, %eax // is num positive and exp >= 6 (number is >= 128.0)? jae HIDDEN_JUMPTARGET (__expl) // (if num is denormal, it is at least >= 64.0) #endif fldt 8(%rsp) /* I added the following ugly construct because expl(+-Inf) resulted in NaN. The ugliness results from the bright minds at Intel. For the i686 the code can be written better. -- drepper@cygnus.com. */ fxam /* Is NaN or +-Inf? */ #ifdef USE_AS_EXPM1L xorb $0x80, %ah cmpl $0xc006, %eax fstsw %ax movb $0x45, %dh jb 4f /* Below -64.0 (may be -NaN or -Inf). */ andb %ah, %dh cmpb $0x01, %dh je 6f /* Is +-NaN, jump. */ jmp 1f /* -large, possibly -Inf. */ 4: /* In range -64.0 to 64.0 (may be +-0 but not NaN or +-Inf). */ /* Test for +-0 as argument. */ andb %ah, %dh cmpb $0x40, %dh je 2f /* Test for arguments that are small but not subnormal. */ movzwl 8+8(%rsp), %eax andl $0x7fff, %eax cmpl $0x3fbf, %eax jge 3f /* Argument's exponent below -64; avoid spurious underflow if normal. */ cmpl $0x0001, %eax jge 2f /* Force underflow and return the argument, to avoid wrong signs of zero results from the code below in some rounding modes. */ fld %st fmul %st fstp %st jmp 2f #else movzwl 8+8(%rsp), %eax andl $0x7fff, %eax cmpl $0x400d, %eax jg 5f cmpl $0x3fbc, %eax jge 3f /* Argument's exponent below -67, result rounds to 1. */ fld1 faddp jmp 2f 5: /* Overflow, underflow or infinity or NaN as argument. */ fstsw %ax movb $0x45, %dh andb %ah, %dh cmpb $0x05, %dh je 1f /* Is +-Inf, jump. */ cmpb $0x01, %dh je 6f /* Is +-NaN, jump. */ /* Overflow or underflow; saturate. */ fstp %st fldt MO(csat) andb $2, %ah jz 3f fchs #endif 3: FLDLOG /* 1 log2(base) */ fmul %st(1), %st /* 1 x log2(base) */ /* Set round-to-nearest temporarily. */ fstcw -4(%rsp) movl $0xf3ff, %edx andl -4(%rsp), %edx movl %edx, -8(%rsp) fldcw -8(%rsp) frndint /* 1 i */ fld %st(1) /* 2 x */ frndint /* 2 xi */ fldcw -4(%rsp) fld %st(1) /* 3 i */ fldt MO(c0) /* 4 c0 */ fld %st(2) /* 5 xi */ fmul %st(1), %st /* 5 c0 xi */ fsubp %st, %st(2) /* 4 f = c0 xi - i */ fld %st(4) /* 5 x */ fsub %st(3), %st /* 5 xf = x - xi */ fmulp %st, %st(1) /* 4 c0 xf */ faddp %st, %st(1) /* 3 f = f + c0 xf */ fldt MO(c1) /* 4 */ fmul %st(4), %st /* 4 c1 * x */ faddp %st, %st(1) /* 3 f = f + c1 * x */ f2xm1 /* 3 2^(fract(x * log2(base))) - 1 */ #ifdef USE_AS_EXPM1L fstp %st(1) /* 2 */ fscale /* 2 scale factor is st(1); base^x - 2^i */ fxch /* 2 i */ fld1 /* 3 1.0 */ fscale /* 3 2^i */ fld1 /* 4 1.0 */ fsubrp %st, %st(1) /* 3 2^i - 1.0 */ fstp %st(1) /* 2 */ faddp %st, %st(1) /* 1 base^x - 1.0 */ #else fld1 /* 4 1.0 */ faddp /* 3 2^(fract(x * log2(base))) */ fstp %st(1) /* 2 */ fscale /* 2 scale factor is st(1); base^x */ fstp %st(1) /* 1 */ LDBL_CHECK_FORCE_UFLOW_NONNEG #endif fstp %st(1) /* 0 */ jmp 2f 1: #ifdef USE_AS_EXPM1L /* For expm1l, only negative sign gets here. */ fstp %st fld1 fchs #else testl $0x200, %eax /* Test sign. */ jz 2f /* If positive, jump. */ fstp %st fldz /* Set result to 0. */ #endif 2: ret 6: /* NaN argument. */ fadd %st ret END(IEEE754_EXPL) #ifdef USE_AS_EXPM1L libm_hidden_def (__expm1l) libm_alias_ldouble (__expm1, expm1) #elif defined USE_AS_EXP10L libm_alias_finite (__ieee754_exp10l, __exp10l) #else libm_alias_finite (__ieee754_expl, __expl) #endif