The cexp() and {,c}{cos,sin}h functions all need to be able to compute

exp(x) scaled down by some factor, and the challenge is doing this
accurately when exp(x) would overflow.  This change replaces all of
the tricks we've been using with common __ldexp_exp() and
__ldexp_cexp() routines that handle all the scaling.

bde plans to improve on this further by moving the guts of exp() into
k_exp.c and handling the scaling in a more direct manner.  But the
current approach is simple and adequate for now.

6 files changed, 210 insertions, 33 deletions

diff --git a/lib/msun/Makefile b/lib/msun/Makefile
index 7542f639457e..f1385f13dc63 100644
--- a/lib/msun/Makefile
+++ b/lib/msun/Makefile
@@ -49,7 +49,7 @@ COMMON_SRCS= b_exp.c b_log.c b_tgamma.c \
 	e_pow.c e_powf.c e_rem_pio2.c \
 	e_rem_pio2f.c e_remainder.c e_remainderf.c e_scalb.c e_scalbf.c \
 	e_sinh.c e_sinhf.c e_sqrt.c e_sqrtf.c fenv.c \
-	k_cos.c k_cosf.c k_rem_pio2.c k_sin.c k_sinf.c \
+	k_cos.c k_cosf.c k_exp.c k_expf.c k_rem_pio2.c k_sin.c k_sinf.c \
 	k_tan.c k_tanf.c \
 	s_asinh.c s_asinhf.c s_atan.c s_atanf.c s_carg.c s_cargf.c s_cargl.c \
 	s_cbrt.c s_cbrtf.c s_ceil.c s_ceilf.c \
diff --git a/lib/msun/src/k_exp.c b/lib/msun/src/k_exp.c
new file mode 100644
index 000000000000..f592f69aa009
--- /dev/null
+++ b/lib/msun/src/k_exp.c
@@ -0,0 +1,108 @@
+/*-
+ * Copyright (c) 2011 David Schultz <das@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 "math.h"
+#include "math_private.h"
+
+static const uint32_t k = 1799;		/* constant for reduction */
+static const double kln2 =  1246.97177782734161156;	/* k * ln2 */
+
+/*
+ * Compute exp(x), scaled to avoid spurious overflow.  An exponent is
+ * returned separately in 'expt'.
+ *
+ * Input:  ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
+ * Output: 2**1023 <= y < 2**1024
+ */
+static double
+__frexp_exp(double x, int *expt)
+{
+	double exp_x;
+	uint32_t hx;
+
+	/*
+	 * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
+	 * minimize |exp(kln2) - 2**k|.  We also scale the exponent of
+	 * exp_x to MAX_EXP so that the result can be multiplied by
+	 * a tiny number without losing accuracy due to denormalization.
+	 */
+	exp_x = exp(x - kln2);
+	GET_HIGH_WORD(hx, exp_x);
+	*expt = (hx >> 20) - (0x3ff + 1023) + k;
+	SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
+	return (exp_x);
+}
+
+/*
+ * __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
+ * They are intended for large arguments (real part >= ln(DBL_MAX))
+ * where care is needed to avoid overflow.
+ *
+ * The present implementation is narrowly tailored for our hyperbolic and
+ * exponential functions.  We assume expt is small (0 or -1), and the caller
+ * has filtered out very large x, for which overflow would be inevitable.
+ */
+
+double
+__ldexp_exp(double x, int expt)
+{
+	double exp_x, scale;
+	int ex_expt;
+
+	exp_x = __frexp_exp(x, &ex_expt);
+	expt += ex_expt;
+	INSERT_WORDS(scale, (0x3ff + expt) << 20, 0);
+	return (exp_x * scale);
+}
+
+double complex
+__ldexp_cexp(double complex z, int expt)
+{
+	double x, y, exp_x, scale1, scale2;
+	int ex_expt, half_expt;
+
+	x = creal(z);
+	y = cimag(z);
+	exp_x = __frexp_exp(x, &ex_expt);
+	expt += ex_expt;
+
+	/*
+	 * Arrange so that scale1 * scale2 == 2**expt.  We use this to
+	 * compensate for scalbn being horrendously slow.
+	 */
+	half_expt = expt / 2;
+	INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
+	half_expt = expt - half_expt;
+	INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);
+
+	return (cpack(cos(y) * exp_x * scale1 * scale2,
+	    sin(y) * exp_x * scale1 * scale2));
+}
diff --git a/lib/msun/src/k_expf.c b/lib/msun/src/k_expf.c
new file mode 100644
index 000000000000..a860b9f74aed
--- /dev/null
+++ b/lib/msun/src/k_expf.c
@@ -0,0 +1,87 @@
+/*-
+ * Copyright (c) 2011 David Schultz <das@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 "math.h"
+#include "math_private.h"
+
+static const uint32_t k = 235;			/* constant for reduction */
+static const float kln2 =  162.88958740F;	/* k * ln2 */
+
+/*
+ * See k_exp.c for details.
+ *
+ * Input:  ln(FLT_MAX) <= x < ln(2 * FLT_MAX / FLT_MIN_DENORM) ~= 192.7
+ * Output: 2**127 <= y < 2**128
+ */
+static float
+__frexp_expf(float x, int *expt)
+{
+	double exp_x;
+	uint32_t hx;
+
+	exp_x = expf(x - kln2);
+	GET_FLOAT_WORD(hx, exp_x);
+	*expt = (hx >> 23) - (0x7f + 127) + k;
+	SET_FLOAT_WORD(exp_x, (hx & 0x7fffff) | ((0x7f + 127) << 23));
+	return (exp_x);
+}
+
+float
+__ldexp_expf(float x, int expt)
+{
+	float exp_x, scale;
+	int ex_expt;
+
+	exp_x = __frexp_expf(x, &ex_expt);
+	expt += ex_expt;
+	SET_FLOAT_WORD(scale, (0x7f + expt) << 23);
+	return (exp_x * scale);
+}
+
+float complex
+__ldexp_cexpf(float complex z, int expt)
+{
+	float x, y, exp_x, scale1, scale2;
+	int ex_expt, half_expt;
+
+	x = crealf(z);
+	y = cimagf(z);
+	exp_x = __frexp_expf(x, &ex_expt);
+	expt += ex_expt;
+
+	half_expt = expt / 2;
+	SET_FLOAT_WORD(scale1, (0x7f + half_expt) << 23);
+	half_expt = expt - half_expt;
+	SET_FLOAT_WORD(scale2, (0x7f + half_expt) << 23);
+
+	return (cpackf(cosf(y) * exp_x * scale1 * scale2,
+	    sinf(y) * exp_x * scale1 * scale2));
+}
diff --git a/lib/msun/src/math_private.h b/lib/msun/src/math_private.h
index 2f8a9db35496..79280e30aa2a 100644
--- a/lib/msun/src/math_private.h
+++ b/lib/msun/src/math_private.h
@@ -397,6 +397,10 @@ int	__ieee754_rem_pio2(double,double*);
 double	__kernel_sin(double,double,int);
 double	__kernel_cos(double,double);
 double	__kernel_tan(double,double,int);
+double	__ldexp_exp(double,int);
+#ifdef _COMPLEX_H
+double complex __ldexp_cexp(double complex,int);
+#endif
 
 /* float precision kernel functions */
 #ifdef INLINE_REM_PIO2F
@@ -415,6 +419,10 @@ float	__kernel_cosdf(double);
 __inline
 #endif
 float	__kernel_tandf(double,int);
+float	__ldexp_expf(float,int);
+#ifdef _COMPLEX_H
+float complex __ldexp_cexpf(float complex,int);
+#endif
 
 /* long double precision kernel functions */
 long double __kernel_sinl(long double, long double, int);
diff --git a/lib/msun/src/s_cexp.c b/lib/msun/src/s_cexp.c
index b907e1d39dd7..abe178f3169f 100644
--- a/lib/msun/src/s_cexp.c
+++ b/lib/msun/src/s_cexp.c
@@ -34,18 +34,13 @@ __FBSDID("$FreeBSD$");
 
 static const uint32_t
 exp_ovfl  = 0x40862e42,			/* high bits of MAX_EXP * ln2 ~= 710 */
-cexp_ovfl = 0x4096b8e4,			/* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
-k         = 1799;			/* constant for reduction */
-
-static const double
-kln2      =  1246.97177782734161156;	/* k * ln2 */
+cexp_ovfl = 0x4096b8e4;			/* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
 
 double complex
 cexp(double complex z)
 {
 	double x, y, exp_x;
 	uint32_t hx, hy, lx, ly;
-	int scale;
 
 	x = creal(z);
 	y = cimag(z);
@@ -77,17 +72,9 @@ cexp(double complex z)
 	if (hx >= exp_ovfl && hx <= cexp_ovfl) {
 		/*
 		 * x is between 709.7 and 1454.3, so we must scale to avoid
-		 * overflow in exp(x).  We use exp(x) = exp(x - kln2) * 2**k,
-		 * carefully chosen to minimize |exp(kln2) - 2**k|.  We also
-		 * scale the exponent of exp(x) to MANT_DIG to avoid loss of
-		 * accuracy due to underflow if sin(y) is tiny.
+		 * overflow in exp(x).
 		 */
-		exp_x = exp(x - kln2);
-		GET_HIGH_WORD(hx, exp_x);
-		SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 52) << 20));
-		scale = (hx >> 20) - (0x3ff + 52) + k;
-		return (cpack(scalbn(cos(y) * exp_x, scale),
-			scalbn(sin(y) * exp_x, scale)));
+		return (__ldexp_cexp(z, 0));
 	} else {
 		/*
 		 * Cases covered here:
diff --git a/lib/msun/src/s_cexpf.c b/lib/msun/src/s_cexpf.c
index 08ec545f7910..0e30d08c80db 100644
--- a/lib/msun/src/s_cexpf.c
+++ b/lib/msun/src/s_cexpf.c
@@ -34,18 +34,13 @@ __FBSDID("$FreeBSD$");
 
 static const uint32_t
 exp_ovfl  = 0x42b17218,		/* MAX_EXP * ln2 ~= 88.722839355 */
-cexp_ovfl = 0x43400074,		/* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
-k         = 235;		/* constant for reduction */
-
-static const float
-kln2      =  162.88958740f;	/* k * ln2 */
+cexp_ovfl = 0x43400074;		/* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
 
 float complex
 cexpf(float complex z)
 {
 	float x, y, exp_x;
 	uint32_t hx, hy;
-	int scale;
 
 	x = crealf(z);
 	y = cimagf(z);
@@ -77,17 +72,9 @@ cexpf(float complex z)
 	if (hx >= exp_ovfl && hx <= cexp_ovfl) {
 		/*
 		 * x is between 88.7 and 192, so we must scale to avoid
-		 * overflow in expf(x).  We use exp(x) = exp(x - kln2) * 2**k,
-		 * carefully chosen to minimize |exp(kln2) - 2**k|.  We also
-		 * scale the exponent of exp(x) to MANT_DIG to avoid loss of
-		 * accuracy due to underflow if sin(y) is tiny.
+		 * overflow in expf(x).
 		 */
-		exp_x = expf(x - kln2);
-		GET_FLOAT_WORD(hx, exp_x);
-		SET_FLOAT_WORD(exp_x, (hx & 0x7fffff) | ((0x7f + 23) << 23));
-		scale = (hx >> 23) - (0x7f + 23) + k;
-		return (cpackf(scalbnf(cosf(y) * exp_x, scale),
-			scalbnf(sinf(y) * exp_x, scale)));
+		return (__ldexp_cexpf(z, 0));
 	} else {
 		/*
 		 * Cases covered here: