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 <i386-math-asm.h>
26#include <libm-alias-finite.h>
27
28#ifdef USE_AS_EXP10L
29# define IEEE754_EXPL __ieee754_exp10l
30# define FLDLOG fldl2t
31#elif defined USE_AS_EXPM1L
32# define IEEE754_EXPL __expm1l
33# define FLDLOG fldl2e
34#else
35# define IEEE754_EXPL __ieee754_expl
36# define FLDLOG fldl2e
37#endif
38
39 .section .rodata.cst16,"aM",@progbits,16
40
41 .p2align 4
42#ifdef USE_AS_EXP10L
43 .type c0,@object
44c0: .byte 0, 0, 0, 0, 0, 0, 0x9a, 0xd4, 0x00, 0x40
45 .byte 0, 0, 0, 0, 0, 0
46 ASM_SIZE_DIRECTIVE(c0)
47 .type c1,@object
48c1: .byte 0x58, 0x92, 0xfc, 0x15, 0x37, 0x9a, 0x97, 0xf0, 0xef, 0x3f
49 .byte 0, 0, 0, 0, 0, 0
50 ASM_SIZE_DIRECTIVE(c1)
51#else
52 .type c0,@object
53c0: .byte 0, 0, 0, 0, 0, 0, 0xaa, 0xb8, 0xff, 0x3f
54 .byte 0, 0, 0, 0, 0, 0
55 ASM_SIZE_DIRECTIVE(c0)
56 .type c1,@object
57c1: .byte 0x20, 0xfa, 0xee, 0xc2, 0x5f, 0x70, 0xa5, 0xec, 0xed, 0x3f
58 .byte 0, 0, 0, 0, 0, 0
59 ASM_SIZE_DIRECTIVE(c1)
60#endif
61#ifndef USE_AS_EXPM1L
62 .type csat,@object
63csat: .byte 0, 0, 0, 0, 0, 0, 0, 0x80, 0x0e, 0x40
64 .byte 0, 0, 0, 0, 0, 0
65 ASM_SIZE_DIRECTIVE(csat)
66DEFINE_LDBL_MIN
67#endif
68
69#ifdef PIC
70# define MO(op) op##@GOTOFF(%ecx)
71#else
72# define MO(op) op
73#endif
74
75 .text
76ENTRY(IEEE754_EXPL)
77#ifdef USE_AS_EXPM1L
78 movzwl 4+8(%esp), %eax
79 xorb $0x80, %ah // invert sign bit (now 1 is "positive")
80 cmpl $0xc006, %eax // is num positive and exp >= 6 (number is >= 128.0)?
81 jae HIDDEN_JUMPTARGET (__expl) // (if num is denormal, it is at least >= 64.0)
82#endif
83 fldt 4(%esp)
84/* I added the following ugly construct because expl(+-Inf) resulted
85 in NaN. The ugliness results from the bright minds at Intel.
86 For the i686 the code can be written better.
87 -- drepper@cygnus.com. */
88 fxam /* Is NaN or +-Inf? */
89#ifdef PIC
90 LOAD_PIC_REG (cx)
91#endif
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 4+8(%esp), %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 4+8(%esp), %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 subl $8, %esp
156 cfi_adjust_cfa_offset (8)
157 fstcw 4(%esp)
158 movl $0xf3ff, %edx
159 andl 4(%esp), %edx
160 movl %edx, (%esp)
161 fldcw (%esp)
162 frndint /* 1 i */
163 fld %st(1) /* 2 x */
164 frndint /* 2 xi */
165 fldcw 4(%esp)
166 addl $8, %esp
167 cfi_adjust_cfa_offset (-8)
168 fld %st(1) /* 3 i */
169 fldt MO(c0) /* 4 c0 */
170 fld %st(2) /* 5 xi */
171 fmul %st(1), %st /* 5 c0 xi */
172 fsubp %st, %st(2) /* 4 f = c0 xi - i */
173 fld %st(4) /* 5 x */
174 fsub %st(3), %st /* 5 xf = x - xi */
175 fmulp %st, %st(1) /* 4 c0 xf */
176 faddp %st, %st(1) /* 3 f = f + c0 xf */
177 fldt MO(c1) /* 4 */
178 fmul %st(4), %st /* 4 c1 * x */
179 faddp %st, %st(1) /* 3 f = f + c1 * x */
180 f2xm1 /* 3 2^(fract(x * log2(base))) - 1 */
181#ifdef USE_AS_EXPM1L
182 fstp %st(1) /* 2 */
183 fscale /* 2 scale factor is st(1); base^x - 2^i */
184 fxch /* 2 i */
185 fld1 /* 3 1.0 */
186 fscale /* 3 2^i */
187 fld1 /* 4 1.0 */
188 fsubrp %st, %st(1) /* 3 2^i - 1.0 */
189 fstp %st(1) /* 2 */
190 faddp %st, %st(1) /* 1 base^x - 1.0 */
191#else
192 fld1 /* 4 1.0 */
193 faddp /* 3 2^(fract(x * log2(base))) */
194 fstp %st(1) /* 2 */
195 fscale /* 2 scale factor is st(1); base^x */
196 fstp %st(1) /* 1 */
197 LDBL_CHECK_FORCE_UFLOW_NONNEG
198#endif
199 fstp %st(1) /* 0 */
200 jmp 2f
2011:
202#ifdef USE_AS_EXPM1L
203 /* For expm1l, only negative sign gets here. */
204 fstp %st
205 fld1
206 fchs
207#else
208 testl $0x200, %eax /* Test sign. */
209 jz 2f /* If positive, jump. */
210 fstp %st
211 fldz /* Set result to 0. */
212#endif
2132: ret
2146: /* NaN argument. */
215 fadd %st
216 ret
217END(IEEE754_EXPL)
218
219#ifdef USE_AS_EXPM1L
220libm_hidden_def (__expm1l)
221libm_alias_ldouble (__expm1, expm1)
222#elif defined USE_AS_EXP10L
223libm_alias_finite (__ieee754_exp10l, __exp10l)
224#else
225libm_alias_finite (__ieee754_expl, __expl)
226#endif
227

source code of glibc/sysdeps/i386/fpu/e_expl.S