1/*
2 * Public domain.
3 *
4 */
5
6/*
7 * The 8087 method for the exponential function is to calculate
8 * exp(x) = 2^(x log2(e))
9 * after separating integer and fractional parts
10 * x log2(e) = i + f, |f| <= .5
11 * 2^i is immediate but f needs to be precise for long double accuracy.
12 * Suppress range reduction error in computing f by the following.
13 * Separate x into integer and fractional parts
14 * x = xi + xf, |xf| <= .5
15 * Separate log2(e) into the sum of an exact number c0 and small part c1.
16 * c0 + c1 = log2(e) to extra precision
17 * Then
18 * f = (c0 xi - i) + c0 xf + c1 x
19 * where c0 xi is exact and so also is (c0 xi - i).
20 * -- moshier@na-net.ornl.gov
21 */
22
23#include <libm-alias-ldouble.h>
24#include <machine/asm.h>
25#include <x86_64-math-asm.h>
26#include <libm-alias-finite.h>
27
28#ifdef USE_AS_EXP10L
29# define IEEE754_EXPL __ieee754_exp10l
30# define EXPL_FINITE __exp10l_finite
31# define FLDLOG fldl2t
32#elif defined USE_AS_EXPM1L
33# define IEEE754_EXPL __expm1l
34# undef EXPL_FINITE
35# define FLDLOG fldl2e
36#else
37# define IEEE754_EXPL __ieee754_expl
38# define EXPL_FINITE __expl_finite
39# define FLDLOG fldl2e
40#endif
41
42 .section .rodata.cst16,"aM",@progbits,16
43
44 .p2align 4
45#ifdef USE_AS_EXP10L
46 .type c0,@object
47c0: .byte 0, 0, 0, 0, 0, 0, 0x9a, 0xd4, 0x00, 0x40
48 .byte 0, 0, 0, 0, 0, 0
49 ASM_SIZE_DIRECTIVE(c0)
50 .type c1,@object
51c1: .byte 0x58, 0x92, 0xfc, 0x15, 0x37, 0x9a, 0x97, 0xf0, 0xef, 0x3f
52 .byte 0, 0, 0, 0, 0, 0
53 ASM_SIZE_DIRECTIVE(c1)
54#else
55 .type c0,@object
56c0: .byte 0, 0, 0, 0, 0, 0, 0xaa, 0xb8, 0xff, 0x3f
57 .byte 0, 0, 0, 0, 0, 0
58 ASM_SIZE_DIRECTIVE(c0)
59 .type c1,@object
60c1: .byte 0x20, 0xfa, 0xee, 0xc2, 0x5f, 0x70, 0xa5, 0xec, 0xed, 0x3f
61 .byte 0, 0, 0, 0, 0, 0
62 ASM_SIZE_DIRECTIVE(c1)
63#endif
64#ifndef USE_AS_EXPM1L
65 .type csat,@object
66csat: .byte 0, 0, 0, 0, 0, 0, 0, 0x80, 0x0e, 0x40
67 .byte 0, 0, 0, 0, 0, 0
68 ASM_SIZE_DIRECTIVE(csat)
69DEFINE_LDBL_MIN
70#endif
71
72#ifdef PIC
73# define MO(op) op##(%rip)
74#else
75# define MO(op) op
76#endif
77
78 .text
79ENTRY(IEEE754_EXPL)
80#ifdef USE_AS_EXPM1L
81 movzwl 8+8(%rsp), %eax
82 xorb $0x80, %ah // invert sign bit (now 1 is "positive")
83 cmpl $0xc006, %eax // is num positive and exp >= 6 (number is >= 128.0)?
84 jae HIDDEN_JUMPTARGET (__expl) // (if num is denormal, it is at least >= 64.0)
85#endif
86 fldt 8(%rsp)
87/* I added the following ugly construct because expl(+-Inf) resulted
88 in NaN. The ugliness results from the bright minds at Intel.
89 For the i686 the code can be written better.
90 -- drepper@cygnus.com. */
91 fxam /* Is NaN or +-Inf? */
92#ifdef USE_AS_EXPM1L
93 xorb $0x80, %ah
94 cmpl $0xc006, %eax
95 fstsw %ax
96 movb $0x45, %dh
97 jb 4f
98
99 /* Below -64.0 (may be -NaN or -Inf). */
100 andb %ah, %dh
101 cmpb $0x01, %dh
102 je 6f /* Is +-NaN, jump. */
103 jmp 1f /* -large, possibly -Inf. */
104
1054: /* In range -64.0 to 64.0 (may be +-0 but not NaN or +-Inf). */
106 /* Test for +-0 as argument. */
107 andb %ah, %dh
108 cmpb $0x40, %dh
109 je 2f
110
111 /* Test for arguments that are small but not subnormal. */
112 movzwl 8+8(%rsp), %eax
113 andl $0x7fff, %eax
114 cmpl $0x3fbf, %eax
115 jge 3f
116 /* Argument's exponent below -64; avoid spurious underflow if
117 normal. */
118 cmpl $0x0001, %eax
119 jge 2f
120 /* Force underflow and return the argument, to avoid wrong signs
121 of zero results from the code below in some rounding modes. */
122 fld %st
123 fmul %st
124 fstp %st
125 jmp 2f
126#else
127 movzwl 8+8(%rsp), %eax
128 andl $0x7fff, %eax
129 cmpl $0x400d, %eax
130 jg 5f
131 cmpl $0x3fbc, %eax
132 jge 3f
133 /* Argument's exponent below -67, result rounds to 1. */
134 fld1
135 faddp
136 jmp 2f
1375: /* Overflow, underflow or infinity or NaN as argument. */
138 fstsw %ax
139 movb $0x45, %dh
140 andb %ah, %dh
141 cmpb $0x05, %dh
142 je 1f /* Is +-Inf, jump. */
143 cmpb $0x01, %dh
144 je 6f /* Is +-NaN, jump. */
145 /* Overflow or underflow; saturate. */
146 fstp %st
147 fldt MO(csat)
148 andb $2, %ah
149 jz 3f
150 fchs
151#endif
1523: FLDLOG /* 1 log2(base) */
153 fmul %st(1), %st /* 1 x log2(base) */
154 /* Set round-to-nearest temporarily. */
155 fstcw -4(%rsp)
156 movl $0xf3ff, %edx
157 andl -4(%rsp), %edx
158 movl %edx, -8(%rsp)
159 fldcw -8(%rsp)
160 frndint /* 1 i */
161 fld %st(1) /* 2 x */
162 frndint /* 2 xi */
163 fldcw -4(%rsp)
164 fld %st(1) /* 3 i */
165 fldt MO(c0) /* 4 c0 */
166 fld %st(2) /* 5 xi */
167 fmul %st(1), %st /* 5 c0 xi */
168 fsubp %st, %st(2) /* 4 f = c0 xi - i */
169 fld %st(4) /* 5 x */
170 fsub %st(3), %st /* 5 xf = x - xi */
171 fmulp %st, %st(1) /* 4 c0 xf */
172 faddp %st, %st(1) /* 3 f = f + c0 xf */
173 fldt MO(c1) /* 4 */
174 fmul %st(4), %st /* 4 c1 * x */
175 faddp %st, %st(1) /* 3 f = f + c1 * x */
176 f2xm1 /* 3 2^(fract(x * log2(base))) - 1 */
177#ifdef USE_AS_EXPM1L
178 fstp %st(1) /* 2 */
179 fscale /* 2 scale factor is st(1); base^x - 2^i */
180 fxch /* 2 i */
181 fld1 /* 3 1.0 */
182 fscale /* 3 2^i */
183 fld1 /* 4 1.0 */
184 fsubrp %st, %st(1) /* 3 2^i - 1.0 */
185 fstp %st(1) /* 2 */
186 faddp %st, %st(1) /* 1 base^x - 1.0 */
187#else
188 fld1 /* 4 1.0 */
189 faddp /* 3 2^(fract(x * log2(base))) */
190 fstp %st(1) /* 2 */
191 fscale /* 2 scale factor is st(1); base^x */
192 fstp %st(1) /* 1 */
193 LDBL_CHECK_FORCE_UFLOW_NONNEG
194#endif
195 fstp %st(1) /* 0 */
196 jmp 2f
1971:
198#ifdef USE_AS_EXPM1L
199 /* For expm1l, only negative sign gets here. */
200 fstp %st
201 fld1
202 fchs
203#else
204 testl $0x200, %eax /* Test sign. */
205 jz 2f /* If positive, jump. */
206 fstp %st
207 fldz /* Set result to 0. */
208#endif
2092: ret
2106: /* NaN argument. */
211 fadd %st
212 ret
213END(IEEE754_EXPL)
214
215#ifdef USE_AS_EXPM1L
216libm_hidden_def (__expm1l)
217libm_alias_ldouble (__expm1, expm1)
218#elif defined USE_AS_EXP10L
219libm_alias_finite (__ieee754_exp10l, __exp10l)
220#else
221libm_alias_finite (__ieee754_expl, __expl)
222#endif
223