Initial import

libc/math/log1p.c

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+/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+/* double log1p(double x)
+ * Return the natural logarithm of 1+x.
+ *
+ * Method :
+ *   1. Argument Reduction: find k and f such that
+ *                      1+x = 2^k * (1+f),
+ *         where  sqrt(2)/2 < 1+f < sqrt(2) .
+ *
+ *      Note. If k=0, then f=x is exact. However, if k!=0, then f
+ *      may not be representable exactly. In that case, a correction
+ *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
+ *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
+ *      and add back the correction term c/u.
+ *      (Note: when x > 2**53, one can simply return log(x))
+ *
+ *   2. Approximation of log(1+f): See log.c
+ *
+ *   3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
+ *
+ * Special cases:
+ *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
+ *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
+ *      log1p(NaN) is that NaN with no signal.
+ *
+ * Accuracy:
+ *      according to an error analysis, the error is always less than
+ *      1 ulp (unit in the last place).
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ *
+ * Note: Assuming log() return accurate answer, the following
+ *       algorithm can be used to compute log1p(x) to within a few ULP:
+ *
+ *              u = 1+x;
+ *              if(u==1.0) return x ; else
+ *                         return log(u)*(x/(u-1.0));
+ *
+ *       See HP-15C Advanced Functions Handbook, p.193.
+ */
+
+#include "libc/math/libm.h"
+
+static const double
+ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
+ln2_lo = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
+Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
+Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
+Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
+Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
+Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
+Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
+Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
+
+double log1p(double x)
+{
+	union {double f; uint64_t i;} u = {x};
+	double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
+	uint32_t hx,hu;
+	int k;
+
+	hx = u.i>>32;
+	k = 1;
+	if (hx < 0x3fda827a || hx>>31) {  /* 1+x < sqrt(2)+ */
+		if (hx >= 0xbff00000) {  /* x <= -1.0 */
+			if (x == -1)
+				return x/0.0; /* log1p(-1) = -inf */
+			return (x-x)/0.0;     /* log1p(x<-1) = NaN */
+		}
+		if (hx<<1 < 0x3ca00000<<1) {  /* |x| < 2**-53 */
+			/* underflow if subnormal */
+			if ((hx&0x7ff00000) == 0)
+				FORCE_EVAL((float)x);
+			return x;
+		}
+		if (hx <= 0xbfd2bec4) {  /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
+			k = 0;
+			c = 0;
+			f = x;
+		}
+	} else if (hx >= 0x7ff00000)
+		return x;
+	if (k) {
+		u.f = 1 + x;
+		hu = u.i>>32;
+		hu += 0x3ff00000 - 0x3fe6a09e;
+		k = (int)(hu>>20) - 0x3ff;
+		/* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
+		if (k < 54) {
+			c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
+			c /= u.f;
+		} else
+			c = 0;
+		/* reduce u into [sqrt(2)/2, sqrt(2)] */
+		hu = (hu&0x000fffff) + 0x3fe6a09e;
+		u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
+		f = u.f - 1;
+	}
+	hfsq = 0.5*f*f;
+	s = f/(2.0+f);
+	z = s*s;
+	w = z*z;
+	t1 = w*(Lg2+w*(Lg4+w*Lg6));
+	t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
+	R = t2 + t1;
+	dk = k;
+	return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
+}