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I've committed the following patch,
Andreas
2001-05-15 Andreas Jaeger <aj@suse.de>
* sysdeps/ieee754/ldbl-128/s_expm1l.c: New file, contributed by
Stephen L Moshier <moshier@mediaone.net>.
* sysdeps/i386/fpu/libm-test-ulps: Adjust for change.
* math/libm-test.inc: Add comment with ToDo.
* sysdeps/i386/fpu/e_expl.c: Rewritten to C and using a more
accurate algorithm. Patch by Stephen L Moshier <moshier@mediaone.net>.
* sysdeps/i386/fpu/e_expl.S: Removed.
============================================================
Index: sysdeps/i386/fpu/libm-test-ulps
--- sysdeps/i386/fpu/libm-test-ulps 2001/05/14 09:24:51 1.28
+++ sysdeps/i386/fpu/libm-test-ulps 2001/05/15 07:54:47
@@ -484,13 +484,13 @@
# ctan
Test "Real part of: ctan (-2 - 3 i) == 0.0037640256415042482 - 1.0032386273536098014 i":
-ildouble: 437
-ldouble: 437
+ildouble: 439
+ldouble: 439
Test "Imaginary part of: ctan (-2 - 3 i) == 0.0037640256415042482 - 1.0032386273536098014 i":
float: 1
ifloat: 1
-ildouble: 1
-ldouble: 1
+ildouble: 2
+ldouble: 2
Test "Real part of: ctan (0.7 + 1.2 i) == 0.1720734197630349001 + 0.9544807059989405538 i":
double: 1
float: 1
@@ -506,13 +506,13 @@
# ctanh
Test "Real part of: ctanh (-2 - 3 i) == -0.9653858790221331242 + 0.0098843750383224937 i":
-ildouble: 2
-ldouble: 2
+ildouble: 5
+ldouble: 5
Test "Imaginary part of: ctanh (-2 - 3 i) == -0.9653858790221331242 + 0.0098843750383224937 i":
float: 1
ifloat: 1
-ildouble: 23
-ldouble: 23
+ildouble: 25
+ldouble: 25
Test "Real part of: ctanh (0 + pi/4 i) == 0.0 + 1.0 i":
Test "Imaginary part of: ctanh (0 + pi/4 i) == 0.0 + 1.0 i":
float: 1
@@ -551,8 +551,8 @@
float: 12
idouble: 24
ifloat: 12
-ldouble: 4
-ildouble: 4
+ldouble: 12
+ildouble: 12
Test "erfc (9) == 0.41370317465138102381e-36":
ldouble: 36
ildouble: 36
============================================================
Index: math/libm-test.inc
--- math/libm-test.inc 2001/05/14 08:14:51 1.33
+++ math/libm-test.inc 2001/05/15 07:54:47
@@ -104,7 +104,12 @@
- Compiler has errors
With e.g. gcc 2.7.2.2 the test for cexp fails because of a compiler error.
- */
+
+
+ To Do: All parameter should be numbers that can be represented as
+ exact floating point values. Currently some values cannot be represented
+ exactly and therefore the result is not the expected result.
+*/
#ifndef _GNU_SOURCE
# define _GNU_SOURCE
============================================================
Index: sysdeps/ieee754/ldbl-128/s_expm1l.c
--- sysdeps/ieee754/ldbl-128/s_expm1l.c created
+++ sysdeps/ieee754/ldbl-128/s_expm1l.c Tue May 15 09:50:36 2001 1.1
@@ -0,0 +1,145 @@
+/* expm1l.c
+ *
+ * Exponential function, minus 1
+ * 128-bit long double precision
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * long double x, y, expm1l();
+ *
+ * y = expm1l( x );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns e (2.71828...) raised to the x power, minus one.
+ *
+ * Range reduction is accomplished by separating the argument
+ * into an integer k and fraction f such that
+ *
+ * x k f
+ * e = 2 e.
+ *
+ * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
+ * in the basic range [-0.5 ln 2, 0.5 ln 2].
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -79,+MAXLOG 100,000 1.7e-34 4.5e-35
+ *
+ */
+
+/* Copyright 2001 by Stephen L. Moshier */
+
+
+#include "math.h"
+#include "math_private.h"
+
+/* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
+ -.5 ln 2 < x < .5 ln 2
+ Theoretical peak relative error = 8.1e-36 */
+
+static long double
+ P0 = 2.943520915569954073888921213330863757240E8L,
+ P1 = -5.722847283900608941516165725053359168840E7L,
+ P2 = 8.944630806357575461578107295909719817253E6L,
+ P3 = -7.212432713558031519943281748462837065308E5L,
+ P4 = 4.578962475841642634225390068461943438441E4L,
+ P5 = -1.716772506388927649032068540558788106762E3L,
+ P6 = 4.401308817383362136048032038528753151144E1L,
+ P7 = -4.888737542888633647784737721812546636240E-1L,
+ Q0 = 1.766112549341972444333352727998584753865E9L,
+ Q1 = -7.848989743695296475743081255027098295771E8L,
+ Q2 = 1.615869009634292424463780387327037251069E8L,
+ Q3 = -2.019684072836541751428967854947019415698E7L,
+ Q4 = 1.682912729190313538934190635536631941751E6L,
+ Q5 = -9.615511549171441430850103489315371768998E4L,
+ Q6 = 3.697714952261803935521187272204485251835E3L,
+ Q7 = -8.802340681794263968892934703309274564037E1L,
+ /* Q8 = 1.000000000000000000000000000000000000000E0 */
+/* C1 + C2 = ln 2 */
+
+ C1 = 6.93145751953125E-1L,
+ C2 = 1.428606820309417232121458176568075500134E-6L,
+/* ln (2^16384 * (1 - 2^-113)) */
+ maxlog = 1.1356523406294143949491931077970764891253E4L,
+/* ln 2^-114 */
+ minarg = -7.9018778583833765273564461846232128760607E1L, big = 2e4932L;
+
+
+long double
+__expm1l (long double x)
+{
+ long double px, qx, xx;
+ int32_t ix, sign;
+ ieee854_long_double_shape_type u;
+ int k;
+
+ /* Overflow. */
+ if (x > maxlog)
+ return (big * big);
+
+ /* Minimum value. */
+ if (x < minarg)
+ return (4.0 / big - 1.0L);
+
+ /* Detect infinity and NaN. */
+ u.value = x;
+ ix = u.parts32.w0;
+ sign = ix & 0x80000000;
+ ix &= 0x7fffffff;
+ if (ix >= 0x7fff0000)
+ {
+ /* Infinity. */
+ if (((ix & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
+ {
+ if (sign)
+ return -1.0L;
+ else
+ return x;
+ }
+ /* NaN. */
+ return (x + x);
+ }
+
+ /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
+ xx = C1 + C2; /* ln 2. */
+ px = __floorl (0.5 + x / xx);
+ k = px;
+ /* remainder times ln 2 */
+ x -= px * C1;
+ x -= px * C2;
+
+ /* Approximate exp(remainder ln 2). */
+ px = (((((((P7 * x
+ + P6) * x
+ + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
+
+ qx = (((((((x
+ + Q7) * x
+ + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
+
+ xx = x * x;
+ qx = x + (0.5 * xx + xx * px / qx);
+
+ /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
+
+ We have qx = exp(remainder ln 2) - 1, so
+ exp(x) - 1 = 2^k (qx + 1) - 1
+ = 2^k qx + 2^k - 1. */
+
+ px = ldexpl (1.0L, k);
+ x = px * qx + (px - 1.0);
+ return x;
+}
+
+weak_alias (__expm1l, expm1l)
+#ifdef NO_LONG_DOUBLE
+strong_alias (__expm1, __expm1l) weak_alias (__expm1, expm1l)
+#endif
============================================================
Index: sysdeps/i386/fpu/e_expl.c
--- sysdeps/i386/fpu/e_expl.c created
+++ sysdeps/i386/fpu/e_expl.c Tue May 15 09:54:41 2001 1.1
@@ -0,0 +1,75 @@
+/*
+ * 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
+ */
+
+static long double c0 = 1.44268798828125L;
+static long double c1 = 7.05260771340735992468e-6L;
+
+long double
+__ieee754_expl (long double x)
+{
+ long double res, t;
+
+/* 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. */
+ asm ("fxam\n\t" /* Is NaN or +-Inf? */
+ "fstsw %%ax\n\t"
+ "movb $0x45, %%dh\n\t"
+ "andb %%ah, %%dh\n\t"
+ "cmpb $0x05, %%dh\n\t"
+ "je 1f\n\t" /* Is +-Inf, jump. */
+ "fldl2e\n\t" /* 1 log2(e) */
+ "fmul %%st(1),%%st\n\t" /* 1 x log2(e) */
+ "frndint\n\t" /* 1 i */
+ "fld %%st(1)\n\t" /* 2 x */
+ "frndint\n\t" /* 2 xi */
+ "fld %%st(1)\n\t" /* 3 i */
+ "fldt c0\n\t" /* 4 c0 */
+ "fld %%st(2)\n\t" /* 5 xi */
+ "fmul %%st(1),%%st\n\t" /* 5 c0 xi */
+ "fsubp %%st,%%st(2)\n\t" /* 4 f = c0 xi - i */
+ "fld %%st(4)\n\t" /* 5 x */
+ "fsub %%st(3),%%st\n\t" /* 5 xf = x - xi */
+ "fmulp %%st,%%st(1)\n\t" /* 4 c0 xf */
+ "faddp %%st,%%st(1)\n\t" /* 3 f = f + c0 xf */
+ "fldt c1\n\t" /* 4 */
+ "fmul %%st(4),%%st\n\t" /* 4 c1 * x */
+ "faddp %%st,%%st(1)\n\t" /* 3 f = f + c1 * x */
+ "f2xm1\n\t" /* 3 2^(fract(x * log2(e))) - 1 */
+ "fld1\n\t" /* 4 1.0 */
+ "faddp\n\t" /* 3 2^(fract(x * log2(e))) */
+ "fstp %%st(1)\n\t" /* 2 */
+ "fscale\n\t" /* 2 scale factor is st(1); e^x */
+ "fstp %%st(1)\n\t" /* 1 */
+ "fstp %%st(1)\n\t" /* 0 */
+ "jmp 2f\n\t"
+ "1:\ttestl $0x200, %%eax\n\t" /* Test sign. */
+ "jz 2f\n\t" /* If positive, jump. */
+ "fstp %%st\n\t"
+ "fldz\n\t" /* Set result to 0. */
+ "2:\t\n"
+ : "=t" (res) : "0" (x) : "ax", "dx");
+ return res;
+}
--
Andreas Jaeger
SuSE Labs aj@suse.de
private aj@arthur.inka.de
http://www.suse.de/~aj
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