Author: das
Date: Thu May 30 04:49:26 2013
New Revision: 251121
URL: http://svnweb.freebsd.org/changeset/base/251121
Log:
I'm happy to finally commit stephen@'s implementations of cacos,
cacosh, casin, casinh, catan, and catanh. Thanks to stephen@ and bde@
for working on these.
Submitted by: stephen@
Reviewed by: bde
Added:
head/lib/msun/man/cacos.3 (contents, props changed)
head/lib/msun/src/catrig.c (contents, props changed)
head/lib/msun/src/catrigf.c (contents, props changed)
Modified:
head/include/complex.h
head/lib/msun/Makefile
head/lib/msun/Symbol.map
head/lib/msun/man/ccos.3
head/lib/msun/man/ccosh.3
head/lib/msun/man/complex.3
Modified: head/include/complex.h
==============================================================================
--- head/include/complex.h Thu May 30 04:47:03 2013 (r251120)
+++ head/include/complex.h Thu May 30 04:49:26 2013 (r251121)
@@ -63,9 +63,21 @@ __BEGIN_DECLS
double cabs(double complex);
float cabsf(float complex);
long double cabsl(long double complex);
+double complex cacos(double complex);
+float complex cacosf(float complex);
+double complex cacosh(double complex);
+float complex cacoshf(float complex);
double carg(double complex);
float cargf(float complex);
long double cargl(long double complex);
+double complex casin(double complex);
+float complex casinf(float complex);
+double complex casinh(double complex);
+float complex casinhf(float complex);
+double complex catan(double complex);
+float complex catanf(float complex);
+double complex catanh(double complex);
+float complex catanhf(float complex);
double complex ccos(double complex);
float complex ccosf(float complex);
double complex ccosh(double complex);
Modified: head/lib/msun/Makefile
==============================================================================
--- head/lib/msun/Makefile Thu May 30 04:47:03 2013 (r251120)
+++ head/lib/msun/Makefile Thu May 30 04:49:26 2013 (r251121)
@@ -105,7 +105,8 @@ COMMON_SRCS+= e_acosl.c e_asinl.c e_atan
.endif
# C99 complex functions
-COMMON_SRCS+= s_ccosh.c s_ccoshf.c s_cexp.c s_cexpf.c \
+COMMON_SRCS+= catrig.c catrigf.c \
+ s_ccosh.c s_ccoshf.c s_cexp.c s_cexpf.c \
s_cimag.c s_cimagf.c s_cimagl.c \
s_conj.c s_conjf.c s_conjl.c \
s_cproj.c s_cprojf.c s_creal.c s_crealf.c s_creall.c \
@@ -126,7 +127,7 @@ SRCS= ${COMMON_SRCS} ${ARCH_SRCS}
INCS+= fenv.h math.h
MAN= acos.3 acosh.3 asin.3 asinh.3 atan.3 atan2.3 atanh.3 \
- ceil.3 ccos.3 ccosh.3 cexp.3 \
+ ceil.3 cacos.3 ccos.3 ccosh.3 cexp.3 \
cimag.3 copysign.3 cos.3 cosh.3 csqrt.3 erf.3 exp.3 fabs.3 fdim.3 \
feclearexcept.3 feenableexcept.3 fegetenv.3 \
fegetround.3 fenv.3 floor.3 \
@@ -144,6 +145,9 @@ MLINKS+=atan.3 atanf.3 atan.3 atanl.3
MLINKS+=atanh.3 atanhf.3
MLINKS+=atan2.3 atan2f.3 atan2.3 atan2l.3 \
atan2.3 carg.3 atan2.3 cargf.3 atan2.3 cargl.3
+MLINKS+=cacos.3 cacosf.3 cacos.3 cacosh.3 cacos.3 cacoshf.3 \
+ cacos.3 casin.3 cacos.3 casinf.3 cacos.3 casinh.3 cacos.3 casinhf.3 \
+ cacos.3 catan.3 cacos.3 catanf.3 cacos.3 catanh.3 cacos.3 catanhf.3
MLINKS+=ccos.3 ccosf.3 ccos.3 csin.3 ccos.3 csinf.3 ccos.3 ctan.3 ccos.3
ctanf.3
MLINKS+=ccosh.3 ccoshf.3 ccosh.3 csinh.3 ccosh.3 csinhf.3 \
ccosh.3 ctanh.3 ccosh.3 ctanhf.3
Modified: head/lib/msun/Symbol.map
==============================================================================
--- head/lib/msun/Symbol.map Thu May 30 04:47:03 2013 (r251120)
+++ head/lib/msun/Symbol.map Thu May 30 04:49:26 2013 (r251121)
@@ -237,6 +237,18 @@ FBSD_1.3 {
fegetround;
fesetround;
fesetenv;
+ cacos;
+ cacosf;
+ cacosh;
+ cacoshf;
+ casin;
+ casinf;
+ casinh;
+ casinhf;
+ catan;
+ catanf;
+ catanh;
+ catanhf;
csin;
csinf;
csinh;
Added: head/lib/msun/man/cacos.3
==============================================================================
--- /dev/null 00:00:00 1970 (empty, because file is newly added)
+++ head/lib/msun/man/cacos.3 Thu May 30 04:49:26 2013 (r251121)
@@ -0,0 +1,128 @@
+.\" Copyright (c) 2013 David Schultz <d...@freebsd.org>
+.\" All rights reserved.
+.\"
+.\" Redistribution and use in source and binary forms, with or without
+.\" modification, are permitted provided that the following conditions
+.\" are met:
+.\" 1. Redistributions of source code must retain the above copyright
+.\" notice, this list of conditions and the following disclaimer.
+.\" 2. Redistributions in binary form must reproduce the above copyright
+.\" notice, this list of conditions and the following disclaimer in the
+.\" documentation and/or other materials provided with the distribution.
+.\"
+.\" THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
+.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+.\" ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
+.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+.\" SUCH DAMAGE.
+.\"
+.\" $FreeBSD$
+.\"
+.Dd May 27, 2013
+.Dt CACOS 3
+.Os
+.Sh NAME
+.Nm cacos ,
+.Nm cacosf ,
+.Nm cacosh ,
+.Nm cacoshf ,
+.Nm casin ,
+.Nm casinf
+.Nm casinh ,
+.Nm casinhf
+.Nm catan ,
+.Nm catanf
+.Nm catanh ,
+.Nm catanhf
+.Nd complex arc trigonometric and hyperbolic functions
+.Sh LIBRARY
+.Lb libm
+.Sh SYNOPSIS
+.In complex.h
+.Ft double complex
+.Fn cacos "double complex z"
+.Ft float complex
+.Fn cacosf "float complex z"
+.Ft double complex
+.Fn cacosh "double complex z"
+.Ft float complex
+.Fn cacoshf "float complex z"
+.Ft double complex
+.Fn casin "double complex z"
+.Ft float complex
+.Fn casinf "float complex z"
+.Ft double complex
+.Fn casinh "double complex z"
+.Ft float complex
+.Fn casinhf "float complex z"
+.Ft double complex
+.Fn catan "double complex z"
+.Ft float complex
+.Fn catanf "float complex z"
+.Ft double complex
+.Fn catanh "double complex z"
+.Ft float complex
+.Fn catanhf "float complex z"
+.Sh DESCRIPTION
+The
+.Fn cacos ,
+.Fn casin ,
+and
+.Fn catan
+functions compute the principal value of the inverse cosine, sine,
+and tangent of the complex number
+.Fa z ,
+respectively.
+The
+.Fn cacosh ,
+.Fn casinh ,
+and
+.Fn catanh
+functions compute the principal value of the inverse hyperbolic
+cosine, sine, and tangent.
+The
+.Fn cacosf ,
+.Fn casinf ,
+.Fn catanf
+.Fn cacoshf ,
+.Fn casinhf ,
+and
+.Fn catanhf
+functions perform the same operations in
+.Fa float
+precision.
+.Pp
+.ie '\*[.T]'utf8'
+. ds Un \[cu]
+.el
+. ds Un U
+.
+There is no universal convention for defining the principal values of
+these functions. The following table gives the branch cuts, and the
+corresponding ranges for the return values, adopted by the C language.
+.Bl -column ".Sy Function" ".Sy (-\*(If*I, -I) \*(Un (I, \*(If*I)" ".Sy
[-\*(Pi/2*I, \*(Pi/2*I]"
+.It Sy Function Ta Sy Branch Cut(s) Ta Sy Range
+.It cacos Ta (-\*(If, -1) \*(Un (1, \*(If) Ta [0, \*(Pi]
+.It casin Ta (-\*(If, -1) \*(Un (1, \*(If) Ta [-\*(Pi/2, \*(Pi/2]
+.It catan Ta (-\*(If*I, -i) \*(Un (I, \*(If*I) Ta [-\*(Pi/2, \*(Pi/2]
+.It cacosh Ta (-\*(If, 1) Ta [-\*(Pi*I, \*(Pi*I]
+.It casinh Ta (-\*(If*I, -i) \*(Un (I, \*(If*I) Ta [-\*(Pi/2*I, \*(Pi/2*I]
+.It catanh Ta (-\*(If, -1) \*(Un (1, \*(If) Ta [-\*(Pi/2*I, \*(Pi/2*I]
+.El
+.Sh SEE ALSO
+.Xr ccos 3 ,
+.Xr ccosh 3 ,
+.Xr complex 3 ,
+.Xr cos 3 ,
+.Xr math 3 ,
+.Xr sin 3 ,
+.Xr tan 3
+.Sh STANDARDS
+These functions conform to
+.St -isoC-99 .
Modified: head/lib/msun/man/ccos.3
==============================================================================
--- head/lib/msun/man/ccos.3 Thu May 30 04:47:03 2013 (r251120)
+++ head/lib/msun/man/ccos.3 Thu May 30 04:49:26 2013 (r251121)
@@ -69,6 +69,7 @@ functions perform the same operations in
.Fa float
precision.
.Sh SEE ALSO
+.Xr cacos 3 ,
.Xr ccosh 3 ,
.Xr complex 3 ,
.Xr cos 3 ,
Modified: head/lib/msun/man/ccosh.3
==============================================================================
--- head/lib/msun/man/ccosh.3 Thu May 30 04:47:03 2013 (r251120)
+++ head/lib/msun/man/ccosh.3 Thu May 30 04:49:26 2013 (r251121)
@@ -69,6 +69,7 @@ functions perform the same operations in
.Fa float
precision.
.Sh SEE ALSO
+.Xr cacosh 3 ,
.Xr ccos 3 ,
.Xr complex 3 ,
.Xr cosh 3 ,
Modified: head/lib/msun/man/complex.3
==============================================================================
--- head/lib/msun/man/complex.3 Thu May 30 04:47:03 2013 (r251120)
+++ head/lib/msun/man/complex.3 Thu May 30 04:49:26 2013 (r251121)
@@ -89,6 +89,12 @@ creal compute the real part
.\" Section 7.3.5-6 of ISO C99 standard
.Ss Trigonometric and Hyperbolic Functions
.Cl
+cacos arc cosine
+cacosh arc hyperbolic cosine
+casin arc sine
+casinh arc hyperbolic sine
+catan arc tangent
+catanh arc hyperbolic tangent
ccos cosine
ccosh hyperbolic cosine
csin sine
@@ -111,20 +117,8 @@ The
functions described here conform to
.St -isoC-99 .
.Sh BUGS
-The inverse trigonometric and hyperbolic functions
-.Fn cacos ,
-.Fn cacosh ,
-.Fn casin ,
-.Fn casinh ,
-.Fn catan ,
-and
-.Fn catanh
-are not implemented.
-.Pp
The logarithmic functions
.Fn clog
-are not implemented.
-.Pp
-The power functions
+and the power functions
.Fn cpow
are not implemented.
Added: head/lib/msun/src/catrig.c
==============================================================================
--- /dev/null 00:00:00 1970 (empty, because file is newly added)
+++ head/lib/msun/src/catrig.c Thu May 30 04:49:26 2013 (r251121)
@@ -0,0 +1,643 @@
+/*-
+ * Copyright (c) 2012 Stephen Montgomery-Smith <step...@freebsd.org>
+ * All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+#include <sys/cdefs.h>
+__FBSDID("$FreeBSD$");
+
+#include <complex.h>
+#include <float.h>
+
+#include "math.h"
+#include "math_private.h"
+
+#undef isinf
+#define isinf(x) (fabs(x) == INFINITY)
+#undef isnan
+#define isnan(x) ((x) != (x))
+#define raise_inexact() do { volatile float junk = 1 + tiny; } while(0)
+#undef signbit
+#define signbit(x) (__builtin_signbit(x))
+
+/* We need that DBL_EPSILON^2/128 is larger than FOUR_SQRT_MIN. */
+static const double
+A_crossover = 10, /* Hull et al suggest 1.5, but 10 works better */
+B_crossover = 0.6417, /* suggested by Hull et al */
+FOUR_SQRT_MIN = 0x1p-509, /* >= 4 * sqrt(DBL_MIN)
*/
+QUARTER_SQRT_MAX = 0x1p509, /* <= sqrt(DBL_MAX) / 4 */
+m_e = 2.7182818284590452e0, /* 0x15bf0a8b145769.0p-51 */
+m_ln2 = 6.9314718055994531e-1, /*
0x162e42fefa39ef.0p-53 */
+pio2_hi = 1.5707963267948966e0, /* 0x1921fb54442d18.0p-52 */
+RECIP_EPSILON = 1 / DBL_EPSILON,
+SQRT_3_EPSILON = 2.5809568279517849e-8, /* 0x1bb67ae8584caa.0p-78 */
+SQRT_6_EPSILON = 3.6500241499888571e-8, /* 0x13988e1409212e.0p-77 */
+SQRT_MIN = 0x1p-511; /* >= sqrt(DBL_MIN) */
+
+static const volatile double
+pio2_lo = 6.1232339957367659e-17; /* 0x11a62633145c07.0p-106 */
+static const volatile float
+tiny = 0x1p-100;
+
+static double complex clog_for_large_values(double complex z);
+
+/*
+ * Testing indicates that all these functions are accurate up to 4 ULP.
+ * The functions casin(h) and cacos(h) are about 2.5 times slower than asinh.
+ * The functions catan(h) are a little under 2 times slower than atanh.
+ *
+ * The code for casinh, casin, cacos, and cacosh comes first. The code is
+ * rather complicated, and the four functions are highly interdependent.
+ *
+ * The code for catanh and catan comes at the end. It is much simpler than
+ * the other functions, and the code for these can be disconnected from the
+ * rest of the code.
+ */
+
+/*
+ * ================================
+ * | casinh, casin, cacos, cacosh |
+ * ================================
+ */
+
+/*
+ * The algorithm is very close to that in "Implementing the complex arcsine
+ * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
+ * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
+ * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
+ * http://dl.acm.org/citation.cfm?id=275324.
+ *
+ * Throughout we use the convention z = x + I*y.
+ *
+ * casinh(z) = sign(x)*log(A+sqrt(A*A-1)) + I*asin(B)
+ * where
+ * A = (|z+I| + |z-I|) / 2
+ * B = (|z+I| - |z-I|) / 2 = y/A
+ *
+ * These formulas become numerically unstable:
+ * (a) for Re(casinh(z)) when z is close to the line segment [-I, I] (that
+ * is, Re(casinh(z)) is close to 0);
+ * (b) for Im(casinh(z)) when z is close to either of the intervals
+ * [I, I*infinity) or (-I*infinity, -I] (that is, |Im(casinh(z))| is
+ * close to PI/2).
+ *
+ * These numerical problems are overcome by defining
+ * f(a, b) = (hypot(a, b) - b) / 2 = a*a / (hypot(a, b) + b) / 2
+ * Then if A < A_crossover, we use
+ * log(A + sqrt(A*A-1)) = log1p((A-1) + sqrt((A-1)*(A+1)))
+ * A-1 = f(x, 1+y) + f(x, 1-y)
+ * and if B > B_crossover, we use
+ * asin(B) = atan2(y, sqrt(A*A - y*y)) = atan2(y, sqrt((A+y)*(A-y)))
+ * A-y = f(x, y+1) + f(x, y-1)
+ * where without loss of generality we have assumed that x and y are
+ * non-negative.
+ *
+ * Much of the difficulty comes because the intermediate computations may
+ * produce overflows or underflows. This is dealt with in the paper by Hull
+ * et al by using exception handling. We do this by detecting when
+ * computations risk underflow or overflow. The hardest part is handling the
+ * underflows when computing f(a, b).
+ *
+ * Note that the function f(a, b) does not appear explicitly in the paper by
+ * Hull et al, but the idea may be found on pages 308 and 309. Introducing the
+ * function f(a, b) allows us to concentrate many of the clever tricks in this
+ * paper into one function.
+ */
+
+/*
+ * Function f(a, b, hypot_a_b) = (hypot(a, b) - b) / 2.
+ * Pass hypot(a, b) as the third argument.
+ */
+static inline double
+f(double a, double b, double hypot_a_b)
+{
+ if (b < 0)
+ return ((hypot_a_b - b) / 2);
+ if (b == 0)
+ return (a / 2);
+ return (a * a / (hypot_a_b + b) / 2);
+}
+
+/*
+ * All the hard work is contained in this function.
+ * x and y are assumed positive or zero, and less than RECIP_EPSILON.
+ * Upon return:
+ * rx = Re(casinh(z)) = -Im(cacos(y + I*x)).
+ * B_is_usable is set to 1 if the value of B is usable.
+ * If B_is_usable is set to 0, sqrt_A2my2 = sqrt(A*A - y*y), and new_y = y.
+ * If returning sqrt_A2my2 has potential to result in an underflow, it is
+ * rescaled, and new_y is similarly rescaled.
+ */
+static inline void
+do_hard_work(double x, double y, double *rx, int *B_is_usable, double *B,
+ double *sqrt_A2my2, double *new_y)
+{
+ double R, S, A; /* A, B, R, and S are as in Hull et al. */
+ double Am1, Amy; /* A-1, A-y. */
+
+ R = hypot(x, y + 1); /* |z+I| */
+ S = hypot(x, y - 1); /* |z-I| */
+
+ /* A = (|z+I| + |z-I|) / 2 */
+ A = (R + S) / 2;
+ /*
+ * Mathematically A >= 1. There is a small chance that this will not
+ * be so because of rounding errors. So we will make certain it is
+ * so.
+ */
+ if (A < 1)
+ A = 1;
+
+ if (A < A_crossover) {
+ /*
+ * Am1 = fp + fm, where fp = f(x, 1+y), and fm = f(x, 1-y).
+ * rx = log1p(Am1 + sqrt(Am1*(A+1)))
+ */
+ if (y == 1 && x < DBL_EPSILON*DBL_EPSILON / 128) {
+ /*
+ * fp is of order x^2, and fm = x/2.
+ * A = 1 (inexactly).
+ */
+ *rx = sqrt(x);
+ } else if (x >= DBL_EPSILON * fabs(y - 1)) {
+ /*
+ * Underflow will not occur because
+ * x >= DBL_EPSILON^2/128 >= FOUR_SQRT_MIN
+ */
+ Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
+ *rx = log1p(Am1 + sqrt(Am1 * (A + 1)));
+ } else if (y < 1) {
+ /*
+ * fp = x*x/(1+y)/4, fm = x*x/(1-y)/4, and
+ * A = 1 (inexactly).
+ */
+ *rx = x / sqrt((1 - y) * (1 + y));
+ } else /* if (y > 1) */ {
+ /*
+ * A-1 = y-1 (inexactly).
+ */
+ *rx = log1p((y - 1) + sqrt((y - 1) * (y + 1)));
+ }
+ } else {
+ *rx = log(A + sqrt(A * A - 1));
+ }
+
+ *new_y = y;
+
+ if (y < FOUR_SQRT_MIN) {
+ /*
+ * Avoid a possible underflow caused by y/A. For casinh this
+ * would be legitimate, but will be picked up by invoking atan2
+ * later on. For cacos this would not be legitimate.
+ */
+ *B_is_usable = 0;
+ *sqrt_A2my2 = A * (2 / DBL_EPSILON);
+ *new_y = y * (2 / DBL_EPSILON);
+ return;
+ }
+
+ /* B = (|z+I| - |z-I|) / 2 = y/A */
+ *B = y / A;
+ *B_is_usable = 1;
+
+ if (*B > B_crossover) {
+ *B_is_usable = 0;
+ /*
+ * Amy = fp + fm, where fp = f(x, y+1), and fm = f(x, y-1).
+ * sqrt_A2my2 = sqrt(Amy*(A+y))
+ */
+ if (y == 1 && x < DBL_EPSILON / 128) {
+ /*
+ * fp is of order x^2, and fm = x/2.
+ * A = 1 (inexactly).
+ */
+ *sqrt_A2my2 = sqrt(x) * sqrt((A + y) / 2);
+ } else if (x >= DBL_EPSILON * fabs(y - 1)) {
+ /*
+ * Underflow will not occur because
+ * x >= DBL_EPSILON/128 >= FOUR_SQRT_MIN
+ * and
+ * x >= DBL_EPSILON^2 >= FOUR_SQRT_MIN
+ */
+ Amy = f(x, y + 1, R) + f(x, y - 1, S);
+ *sqrt_A2my2 = sqrt(Amy * (A + y));
+ } else if (y > 1) {
+ /*
+ * fp = x*x/(y+1)/4, fm = x*x/(y-1)/4, and
+ * A = y (inexactly).
+ *
+ * y < RECIP_EPSILON. So the following
+ * scaling should avoid any underflow problems.
+ */
+ *sqrt_A2my2 = x * (4 / DBL_EPSILON / DBL_EPSILON) * y /
+ sqrt((y + 1) * (y - 1));
+ *new_y = y * (4 / DBL_EPSILON / DBL_EPSILON);
+ } else /* if (y < 1) */ {
+ /*
+ * fm = 1-y >= DBL_EPSILON, fp is of order x^2, and
+ * A = 1 (inexactly).
+ */
+ *sqrt_A2my2 = sqrt((1 - y) * (1 + y));
+ }
+ }
+}
+
+/*
+ * casinh(z) = z + O(z^3) as z -> 0
+ *
+ * casinh(z) = sign(x)*clog(sign(x)*z) + O(1/z^2) as z -> infinity
+ * The above formula works for the imaginary part as well, because
+ * Im(casinh(z)) = sign(x)*atan2(sign(x)*y, fabs(x)) + O(y/z^3)
+ * as z -> infinity, uniformly in y
+ */
+double complex
+casinh(double complex z)
+{
+ double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
+ int B_is_usable;
+ double complex w;
+
+ x = creal(z);
+ y = cimag(z);
+ ax = fabs(x);
+ ay = fabs(y);
+
+ if (isnan(x) || isnan(y)) {
+ /* casinh(+-Inf + I*NaN) = +-Inf + I*NaN */
+ if (isinf(x))
+ return (cpack(x, y + y));
+ /* casinh(NaN + I*+-Inf) = opt(+-)Inf + I*NaN */
+ if (isinf(y))
+ return (cpack(y, x + x));
+ /* casinh(NaN + I*0) = NaN + I*0 */
+ if (y == 0)
+ return (cpack(x + x, y));
+ /*
+ * All other cases involving NaN return NaN + I*NaN.
+ * C99 leaves it optional whether to raise invalid if one of
+ * the arguments is not NaN, so we opt not to raise it.
+ */
+ /* Bruce Evans tells me this is the way to do this: */
+ return (cpack(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
+ }
+
+ if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
+ /* clog...() will raise inexact unless x or y is infinite. */
+ if (signbit(x) == 0)
+ w = clog_for_large_values(z) + m_ln2;
+ else
+ w = clog_for_large_values(-z) + m_ln2;
+ return (cpack(copysign(creal(w), x), copysign(cimag(w), y)));
+ }
+
+ /* Avoid spuriously raising inexact for z = 0. */
+ if (x == 0 && y == 0)
+ return (z);
+
+ /* All remaining cases are inexact. */
+ raise_inexact();
+
+ if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
+ return (z);
+
+ do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
+ if (B_is_usable)
+ ry = asin(B);
+ else
+ ry = atan2(new_y, sqrt_A2my2);
+ return (cpack(copysign(rx, x), copysign(ry, y)));
+}
+
+/*
+ * casin(z) = reverse(casinh(reverse(z)))
+ * where reverse(x + I*y) = y + I*x = I*conj(z).
+ */
+double complex
+casin(double complex z)
+{
+ double complex w = casinh(cpack(cimag(z), creal(z)));
+ return (cpack(cimag(w), creal(w)));
+}
+
+/*
+ * cacos(z) = PI/2 - casin(z)
+ * but do the computation carefully so cacos(z) is accurate when z is
+ * close to 1.
+ *
+ * cacos(z) = PI/2 - z + O(z^3) as z -> 0
+ *
+ * cacos(z) = -sign(y)*I*clog(z) + O(1/z^2) as z -> infinity
+ * The above formula works for the real part as well, because
+ * Re(cacos(z)) = atan2(fabs(y), x) + O(y/z^3)
+ * as z -> infinity, uniformly in y
+ */
+double complex
+cacos(double complex z)
+{
+ double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
+ int sx, sy;
+ int B_is_usable;
+ double complex w;
+
+ x = creal(z);
+ y = cimag(z);
+ sx = signbit(x);
+ sy = signbit(y);
+ ax = fabs(x);
+ ay = fabs(y);
+
+ if (isnan(x) || isnan(y)) {
+ /* cacos(+-Inf + I*NaN) = NaN + I*opt(-)Inf */
+ if (isinf(x))
+ return (cpack(y + y, -INFINITY));
+ /* cacos(NaN + I*+-Inf) = NaN + I*-+Inf */
+ if (isinf(y))
+ return (cpack(x + x, -y));
+ /* cacos(0 + I*NaN) = PI/2 + I*NaN with inexact */
+ if (x == 0)
+ return (cpack(pio2_hi + pio2_lo, y + y));
+ /*
+ * All other cases involving NaN return NaN + I*NaN.
+ * C99 leaves it optional whether to raise invalid if one of
+ * the arguments is not NaN, so we opt not to raise it.
+ */
+ return (cpack(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
+ }
+
+ if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
+ /* clog...() will raise inexact unless x or y is infinite. */
+ w = clog_for_large_values(z);
+ rx = fabs(cimag(w));
+ ry = creal(w) + m_ln2;
+ if (sy == 0)
+ ry = -ry;
+ return (cpack(rx, ry));
+ }
+
+ /* Avoid spuriously raising inexact for z = 1. */
+ if (x == 1 && y == 0)
+ return (cpack(0, -y));
+
+ /* All remaining cases are inexact. */
+ raise_inexact();
+
+ if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON/4)
+ return (cpack(pio2_hi - (x - pio2_lo), -y));
+
+ do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
+ if (B_is_usable) {
+ if (sx==0)
+ rx = acos(B);
+ else
+ rx = acos(-B);
+ } else {
+ if (sx==0)
+ rx = atan2(sqrt_A2mx2, new_x);
+ else
+ rx = atan2(sqrt_A2mx2, -new_x);
+ }
+ if (sy == 0)
+ ry = -ry;
+ return (cpack(rx, ry));
+}
+
+/*
+ * cacosh(z) = I*cacos(z) or -I*cacos(z)
+ * where the sign is chosen so Re(cacosh(z)) >= 0.
+ */
+double complex
+cacosh(double complex z)
+{
+ double complex w;
+ double rx, ry;
+
+ w = cacos(z);
+ rx = creal(w);
+ ry = cimag(w);
+ /* cacosh(NaN + I*NaN) = NaN + I*NaN */
+ if (isnan(rx) && isnan(ry))
+ return (cpack(ry, rx));
+ /* cacosh(NaN + I*+-Inf) = +Inf + I*NaN */
+ /* cacosh(+-Inf + I*NaN) = +Inf + I*NaN */
+ if (isnan(rx))
+ return (cpack(fabs(ry), rx));
+ /* cacosh(0 + I*NaN) = NaN + I*NaN */
+ if (isnan(ry))
+ return (cpack(ry, ry));
+ return (cpack(fabs(ry), copysign(rx, cimag(z))));
+}
+
+/*
+ * Optimized version of clog() for |z| finite and larger than ~RECIP_EPSILON.
+ */
+static double complex
+clog_for_large_values(double complex z)
+{
+ double x, y;
+ double ax, ay, t;
+
+ x = creal(z);
+ y = cimag(z);
+ ax = fabs(x);
+ ay = fabs(y);
+ if (ax < ay) {
+ t = ax;
+ ax = ay;
+ ay = t;
+ }
+
+ /*
+ * Avoid overflow in hypot() when x and y are both very large.
+ * Divide x and y by E, and then add 1 to the logarithm. This depends
+ * on E being larger than sqrt(2).
+ * Dividing by E causes an insignificant loss of accuracy; however
+ * this method is still poor since it is uneccessarily slow.
+ */
+ if (ax > DBL_MAX / 2)
+ return (cpack(log(hypot(x / m_e, y / m_e)) + 1, atan2(y, x)));
+
+ /*
+ * Avoid overflow when x or y is large. Avoid underflow when x or
+ * y is small.
+ */
+ if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
+ return (cpack(log(hypot(x, y)), atan2(y, x)));
+
+ return (cpack(log(ax * ax + ay * ay) / 2, atan2(y, x)));
+}
+
+/*
+ *=============================================================================
+ */
+
+/*
+ * =================
+ * | catanh, catan |
+ * =================
+ */
+
+/*
+ * sum_squares(x,y) = x*x + y*y (or just x*x if y*y would underflow).
+ * Assumes x*x and y*y will not overflow.
+ * Assumes x and y are finite.
+ * Assumes y is non-negative.
+ * Assumes fabs(x) >= DBL_EPSILON.
+ */
+static inline double
+sum_squares(double x, double y)
+{
+
+ /* Avoid underflow when y is small. */
+ if (y < SQRT_MIN)
+ return (x * x);
+ return (x * x + y * y);
+}
+
+/*
+ * real_part_reciprocal(x, y) = Re(1/(x+I*y)) = x/(x*x + y*y).
+ * Assumes x and y are not NaN, and one of x and y is larger than
+ * RECIP_EPSILON. We avoid unwarranted underflow. It is important to not use
+ * the code creal(1/z), because the imaginary part may produce an unwanted
+ * underflow.
+ * This is only called in a context where inexact is always raised before
+ * the call, so no effort is made to avoid or force inexact.
+ */
+static inline double
+real_part_reciprocal(double x, double y)
+{
+ double scale;
+ uint32_t hx, hy;
+ int32_t ix, iy;
+
+ /*
+ * This code is inspired by the C99 document n1124.pdf, Section G.5.1,
+ * example 2.
+ */
+ GET_HIGH_WORD(hx, x);
+ ix = hx & 0x7ff00000;
+ GET_HIGH_WORD(hy, y);
+ iy = hy & 0x7ff00000;
+#define BIAS (DBL_MAX_EXP - 1)
+/* XXX more guard digits are useful iff there is extra precision. */
+#define CUTOFF (DBL_MANT_DIG / 2 + 1) /* just half or 1 guard digit */
+ if (ix - iy >= CUTOFF << 20 || isinf(x))
+ return (1 / x); /* +-Inf -> +-0 is special */
+ if (iy - ix >= CUTOFF << 20)
+ return (x / y / y); /* should avoid double div, but hard */
+ if (ix <= (BIAS + DBL_MAX_EXP / 2 - CUTOFF) << 20)
+ return (x / (x * x + y * y));
+ scale = 1;
+ SET_HIGH_WORD(scale, 0x7ff00000 - ix); /* 2**(1-ilogb(x)) */
+ x *= scale;
+ y *= scale;
+ return (x / (x * x + y * y) * scale);
+}
+
+/*
+ * catanh(z) = log((1+z)/(1-z)) / 2
+ * = log1p(4*x / |z-1|^2) / 4
+ * + I * atan2(2*y, (1-x)*(1+x)-y*y) / 2
+ *
+ * catanh(z) = z + O(z^3) as z -> 0
+ *
+ * catanh(z) = 1/z + sign(y)*I*PI/2 + O(1/z^3) as z -> infinity
+ * The above formula works for the real part as well, because
+ * Re(catanh(z)) = x/|z|^2 + O(x/z^4)
+ * as z -> infinity, uniformly in x
+ */
+double complex
+catanh(double complex z)
+{
+ double x, y, ax, ay, rx, ry;
+
+ x = creal(z);
+ y = cimag(z);
+ ax = fabs(x);
+ ay = fabs(y);
+
+ /* This helps handle many cases. */
+ if (y == 0 && ax <= 1)
+ return (cpack(atanh(x), y));
+
+ /* To ensure the same accuracy as atan(), and to filter out z = 0. */
+ if (x == 0)
+ return (cpack(x, atan(y)));
+
+ if (isnan(x) || isnan(y)) {
+ /* catanh(+-Inf + I*NaN) = +-0 + I*NaN */
+ if (isinf(x))
+ return (cpack(copysign(0, x), y + y));
+ /* catanh(NaN + I*+-Inf) = sign(NaN)0 + I*+-PI/2 */
+ if (isinf(y)) {
+ return (cpack(copysign(0, x),
+ copysign(pio2_hi + pio2_lo, y)));
+ }
+ /*
+ * All other cases involving NaN return NaN + I*NaN.
+ * C99 leaves it optional whether to raise invalid if one of
+ * the arguments is not NaN, so we opt not to raise it.
+ */
+ return (cpack(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
+ }
+
+ if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
+ return (cpack(real_part_reciprocal(x, y),
+ copysign(pio2_hi + pio2_lo, y)));
+ }
+
+ if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
+ /*
+ * z = 0 was filtered out above. All other cases must raise
+ * inexact, but this is the only only that needs to do it
+ * explicitly.
+ */
+ raise_inexact();
+ return (z);
+ }
+
+ if (ax == 1 && ay < DBL_EPSILON)
+ rx = (log(ay) - m_ln2) / -2;
+ else
+ rx = log1p(4 * ax / sum_squares(ax - 1, ay)) / 4;
+
+ if (ax == 1)
+ ry = atan2(2, -ay) / 2;
+ else if (ay < DBL_EPSILON)
+ ry = atan2(2 * ay, (1 - ax) * (1 + ax)) / 2;
+ else
+ ry = atan2(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
+
+ return (cpack(copysign(rx, x), copysign(ry, y)));
+}
+
+/*
+ * catan(z) = reverse(catanh(reverse(z)))
+ * where reverse(x + I*y) = y + I*x = I*conj(z).
+ */
+double complex
+catan(double complex z)
+{
+ double complex w = catanh(cpack(cimag(z), creal(z)));
+ return (cpack(cimag(w), creal(w)));
+}
Added: head/lib/msun/src/catrigf.c
==============================================================================
--- /dev/null 00:00:00 1970 (empty, because file is newly added)
+++ head/lib/msun/src/catrigf.c Thu May 30 04:49:26 2013 (r251121)
@@ -0,0 +1,388 @@
+/*-
+ * Copyright (c) 2012 Stephen Montgomery-Smith <step...@freebsd.org>
+ * All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+/*
+ * The algorithm is very close to that in "Implementing the complex arcsine
+ * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
+ * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
+ * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
+ * http://dl.acm.org/citation.cfm?id=275324.
+ *
+ * The code for catrig.c contains complete comments.
+ */
+
+#include <sys/cdefs.h>
+__FBSDID("$FreeBSD$");
+
+#include <complex.h>
+#include <float.h>
+
+#include "math.h"
+#include "math_private.h"
+
+#undef isinf
+#define isinf(x) (fabsf(x) == INFINITY)
+#undef isnan
+#define isnan(x) ((x) != (x))
+#define raise_inexact() do { volatile float junk = 1 + tiny; } while(0)
+#undef signbit
+#define signbit(x) (__builtin_signbitf(x))
+
+static const float
+A_crossover = 10,
+B_crossover = 0.6417,
+FOUR_SQRT_MIN = 0x1p-61,
+QUARTER_SQRT_MAX = 0x1p61,
+m_e = 2.7182818285e0, /* 0xadf854.0p-22 */
+m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */
+pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */
+RECIP_EPSILON = 1 / FLT_EPSILON,
+SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */
+SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */
+SQRT_MIN = 0x1p-63;
+
+static const volatile float
+pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */
+tiny = 0x1p-100;
+
+static float complex clog_for_large_values(float complex z);
+
+static inline float
+f(float a, float b, float hypot_a_b)
+{
+ if (b < 0)
+ return ((hypot_a_b - b) / 2);
+ if (b == 0)
+ return (a / 2);
+ return (a * a / (hypot_a_b + b) / 2);
+}
+
+static inline void
+do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
+ float *sqrt_A2my2, float *new_y)
+{
+ float R, S, A;
*** DIFF OUTPUT TRUNCATED AT 1000 LINES ***
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