1/*
2 * Written by J.T. Conklin <jtc@netbsd.org>.
3 * Public domain.
4 *
5 * Adapted for `long double' by Ulrich Drepper <drepper@cygnus.com>.
6 */
7
8/*
9 * The 8087 method for the exponential function is to calculate
10 * exp(x) = 2^(x log2(e))
11 * after separating integer and fractional parts
12 * x log2(e) = i + f, |f| <= .5
13 * 2^i is immediate but f needs to be precise for long double accuracy.
14 * Suppress range reduction error in computing f by the following.
15 * Separate x into integer and fractional parts
16 * x = xi + xf, |xf| <= .5
17 * Separate log2(e) into the sum of an exact number c0 and small part c1.
18 * c0 + c1 = log2(e) to extra precision
19 * Then
20 * f = (c0 xi - i) + c0 xf + c1 x
21 * where c0 xi is exact and so also is (c0 xi - i).
22 * -- moshier@na-net.ornl.gov
23 */
24
25#include <libm-alias-ldouble.h>
26#include <machine/asm.h>
27#include <x86_64-math-asm.h>
28#include <libm-alias-finite.h>
29
30#ifdef USE_AS_EXP10L
31# define IEEE754_EXPL __ieee754_exp10l
32# define EXPL_FINITE __exp10l_finite
33# define FLDLOG fldl2t
34#elif defined USE_AS_EXPM1L
35# define IEEE754_EXPL __expm1l
36# undef EXPL_FINITE
37# define FLDLOG fldl2e
38#else
39# define IEEE754_EXPL __ieee754_expl
40# define EXPL_FINITE __expl_finite
41# define FLDLOG fldl2e
42#endif
43
44 .section .rodata.cst16,"aM",@progbits,16
45
46 .p2align 4
47#ifdef USE_AS_EXP10L
48 .type c0,@object
49c0: .byte 0, 0, 0, 0, 0, 0, 0x9a, 0xd4, 0x00, 0x40
50 .byte 0, 0, 0, 0, 0, 0
51 ASM_SIZE_DIRECTIVE(c0)
52 .type c1,@object
53c1: .byte 0x58, 0x92, 0xfc, 0x15, 0x37, 0x9a, 0x97, 0xf0, 0xef, 0x3f
54 .byte 0, 0, 0, 0, 0, 0
55 ASM_SIZE_DIRECTIVE(c1)
56#else
57 .type c0,@object
58c0: .byte 0, 0, 0, 0, 0, 0, 0xaa, 0xb8, 0xff, 0x3f
59 .byte 0, 0, 0, 0, 0, 0
60 ASM_SIZE_DIRECTIVE(c0)
61 .type c1,@object
62c1: .byte 0x20, 0xfa, 0xee, 0xc2, 0x5f, 0x70, 0xa5, 0xec, 0xed, 0x3f
63 .byte 0, 0, 0, 0, 0, 0
64 ASM_SIZE_DIRECTIVE(c1)
65#endif
66#ifndef USE_AS_EXPM1L
67 .type csat,@object
68csat: .byte 0, 0, 0, 0, 0, 0, 0, 0x80, 0x0e, 0x40
69 .byte 0, 0, 0, 0, 0, 0
70 ASM_SIZE_DIRECTIVE(csat)
71DEFINE_LDBL_MIN
72#endif
73
74#ifdef PIC
75# define MO(op) op##(%rip)
76#else
77# define MO(op) op
78#endif
79
80 .text
81ENTRY(IEEE754_EXPL)
82#ifdef USE_AS_EXPM1L
83 movzwl 8+8(%rsp), %eax
84 xorb $0x80, %ah // invert sign bit (now 1 is "positive")
85 cmpl $0xc006, %eax // is num positive and exp >= 6 (number is >= 128.0)?
86 jae HIDDEN_JUMPTARGET (__expl) // (if num is denormal, it is at least >= 64.0)
87#endif
88 fldt 8(%rsp)
89/* I added the following ugly construct because expl(+-Inf) resulted
90 in NaN. The ugliness results from the bright minds at Intel.
91 For the i686 the code can be written better.
92 -- drepper@cygnus.com. */
93 fxam /* Is NaN or +-Inf? */
94#ifdef USE_AS_EXPM1L
95 xorb $0x80, %ah
96 cmpl $0xc006, %eax
97 fstsw %ax
98 movb $0x45, %dh
99 jb 4f
100
101 /* Below -64.0 (may be -NaN or -Inf). */
102 andb %ah, %dh
103 cmpb $0x01, %dh
104 je 6f /* Is +-NaN, jump. */
105 jmp 1f /* -large, possibly -Inf. */
106
1074: /* In range -64.0 to 64.0 (may be +-0 but not NaN or +-Inf). */
108 /* Test for +-0 as argument. */
109 andb %ah, %dh
110 cmpb $0x40, %dh
111 je 2f
112
113 /* Test for arguments that are small but not subnormal. */
114 movzwl 8+8(%rsp), %eax
115 andl $0x7fff, %eax
116 cmpl $0x3fbf, %eax
117 jge 3f
118 /* Argument's exponent below -64; avoid spurious underflow if
119 normal. */
120 cmpl $0x0001, %eax
121 jge 2f
122 /* Force underflow and return the argument, to avoid wrong signs
123 of zero results from the code below in some rounding modes. */
124 fld %st
125 fmul %st
126 fstp %st
127 jmp 2f
128#else
129 movzwl 8+8(%rsp), %eax
130 andl $0x7fff, %eax
131 cmpl $0x400d, %eax
132 jg 5f
133 cmpl $0x3fbc, %eax
134 jge 3f
135 /* Argument's exponent below -67, result rounds to 1. */
136 fld1
137 faddp
138 jmp 2f
1395: /* Overflow, underflow or infinity or NaN as argument. */
140 fstsw %ax
141 movb $0x45, %dh
142 andb %ah, %dh
143 cmpb $0x05, %dh
144 je 1f /* Is +-Inf, jump. */
145 cmpb $0x01, %dh
146 je 6f /* Is +-NaN, jump. */
147 /* Overflow or underflow; saturate. */
148 fstp %st
149 fldt MO(csat)
150 andb $2, %ah
151 jz 3f
152 fchs
153#endif
1543: FLDLOG /* 1 log2(base) */
155 fmul %st(1), %st /* 1 x log2(base) */
156 /* Set round-to-nearest temporarily. */
157 fstcw -4(%rsp)
158 movl $0xf3ff, %edx
159 andl -4(%rsp), %edx
160 movl %edx, -8(%rsp)
161 fldcw -8(%rsp)
162 frndint /* 1 i */
163 fld %st(1) /* 2 x */
164 frndint /* 2 xi */
165 fldcw -4(%rsp)
166 fld %st(1) /* 3 i */
167 fldt MO(c0) /* 4 c0 */
168 fld %st(2) /* 5 xi */
169 fmul %st(1), %st /* 5 c0 xi */
170 fsubp %st, %st(2) /* 4 f = c0 xi - i */
171 fld %st(4) /* 5 x */
172 fsub %st(3), %st /* 5 xf = x - xi */
173 fmulp %st, %st(1) /* 4 c0 xf */
174 faddp %st, %st(1) /* 3 f = f + c0 xf */
175 fldt MO(c1) /* 4 */
176 fmul %st(4), %st /* 4 c1 * x */
177 faddp %st, %st(1) /* 3 f = f + c1 * x */
178 f2xm1 /* 3 2^(fract(x * log2(base))) - 1 */
179#ifdef USE_AS_EXPM1L
180 fstp %st(1) /* 2 */
181 fscale /* 2 scale factor is st(1); base^x - 2^i */
182 fxch /* 2 i */
183 fld1 /* 3 1.0 */
184 fscale /* 3 2^i */
185 fld1 /* 4 1.0 */
186 fsubrp %st, %st(1) /* 3 2^i - 1.0 */
187 fstp %st(1) /* 2 */
188 faddp %st, %st(1) /* 1 base^x - 1.0 */
189#else
190 fld1 /* 4 1.0 */
191 faddp /* 3 2^(fract(x * log2(base))) */
192 fstp %st(1) /* 2 */
193 fscale /* 2 scale factor is st(1); base^x */
194 fstp %st(1) /* 1 */
195 LDBL_CHECK_FORCE_UFLOW_NONNEG
196#endif
197 fstp %st(1) /* 0 */
198 jmp 2f
1991:
200#ifdef USE_AS_EXPM1L
201 /* For expm1l, only negative sign gets here. */
202 fstp %st
203 fld1
204 fchs
205#else
206 testl $0x200, %eax /* Test sign. */
207 jz 2f /* If positive, jump. */
208 fstp %st
209 fldz /* Set result to 0. */
210#endif
2112: ret
2126: /* NaN argument. */
213 fadd %st
214 ret
215END(IEEE754_EXPL)
216
217#ifdef USE_AS_EXPM1L
218libm_hidden_def (__expm1l)
219libm_alias_ldouble (__expm1, expm1)
220#elif defined USE_AS_EXP10L
221libm_alias_finite (__ieee754_exp10l, __exp10l)
222#else
223libm_alias_finite (__ieee754_expl, __expl)
224#endif
225