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