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.

1 files changed, 108 insertions, 0 deletions

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));
+}