first commit of the new libm!

thanks to the hard work of Szabolcs Nagy (nsz), identifying the best
(from correctness and license standpoint) implementations from freebsd
and openbsd and cleaning them up! musl should now fully support c99
float and long double math functions, and has near-complete complex
math support. tgmath should also work (fully on gcc-compatible
compilers, and mostly on any c99 compiler).

based largely on commit 0376d44a890fea261506f1fc63833e7a686dca19 from
nsz's libm git repo, with some additions (dummy versions of a few
missing long double complex functions, etc.) by me.

various cleanups still need to be made, including re-adding (if
they're correct) some asm functions that were dropped.

1 files changed, 220 insertions, 0 deletions

diff --git a/src/math/expm1.c b/src/math/expm1.c
new file mode 100644
index 00000000..ffa82264
--- /dev/null
+++ b/src/math/expm1.c
@@ -0,0 +1,220 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.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.
+ * ====================================================
+ */
+/* expm1(x)
+ * Returns exp(x)-1, the exponential of x minus 1.
+ *
+ * Method
+ *   1. Argument reduction:
+ *      Given x, find r and integer k such that
+ *
+ *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
+ *
+ *      Here a correction term c will be computed to compensate
+ *      the error in r when rounded to a floating-point number.
+ *
+ *   2. Approximating expm1(r) by a special rational function on
+ *      the interval [0,0.34658]:
+ *      Since
+ *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
+ *      we define R1(r*r) by
+ *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
+ *      That is,
+ *          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
+ *                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
+ *                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
+ *      We use a special Reme algorithm on [0,0.347] to generate
+ *      a polynomial of degree 5 in r*r to approximate R1. The
+ *      maximum error of this polynomial approximation is bounded
+ *      by 2**-61. In other words,
+ *          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
+ *      where   Q1  =  -1.6666666666666567384E-2,
+ *              Q2  =   3.9682539681370365873E-4,
+ *              Q3  =  -9.9206344733435987357E-6,
+ *              Q4  =   2.5051361420808517002E-7,
+ *              Q5  =  -6.2843505682382617102E-9;
+ *              z   =  r*r,
+ *      with error bounded by
+ *          |                  5           |     -61
+ *          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
+ *          |                              |
+ *
+ *      expm1(r) = exp(r)-1 is then computed by the following
+ *      specific way which minimize the accumulation rounding error:
+ *                             2     3
+ *                            r     r    [ 3 - (R1 + R1*r/2)  ]
+ *            expm1(r) = r + --- + --- * [--------------------]
+ *                            2     2    [ 6 - r*(3 - R1*r/2) ]
+ *
+ *      To compensate the error in the argument reduction, we use
+ *              expm1(r+c) = expm1(r) + c + expm1(r)*c
+ *                         ~ expm1(r) + c + r*c
+ *      Thus c+r*c will be added in as the correction terms for
+ *      expm1(r+c). Now rearrange the term to avoid optimization
+ *      screw up:
+ *                      (      2                                    2 )
+ *                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
+ *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
+ *                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
+ *                      (                                             )
+ *
+ *                 = r - E
+ *   3. Scale back to obtain expm1(x):
+ *      From step 1, we have
+ *         expm1(x) = either 2^k*[expm1(r)+1] - 1
+ *                  = or     2^k*[expm1(r) + (1-2^-k)]
+ *   4. Implementation notes:
+ *      (A). To save one multiplication, we scale the coefficient Qi
+ *           to Qi*2^i, and replace z by (x^2)/2.
+ *      (B). To achieve maximum accuracy, we compute expm1(x) by
+ *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
+ *        (ii)  if k=0, return r-E
+ *        (iii) if k=-1, return 0.5*(r-E)-0.5
+ *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
+ *                     else          return  1.0+2.0*(r-E);
+ *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
+ *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
+ *        (vii) return 2^k(1-((E+2^-k)-r))
+ *
+ * Special cases:
+ *      expm1(INF) is INF, expm1(NaN) is NaN;
+ *      expm1(-INF) is -1, and
+ *      for finite argument, only expm1(0)=0 is exact.
+ *
+ * Accuracy:
+ *      according to an error analysis, the error is always less than
+ *      1 ulp (unit in the last place).
+ *
+ * Misc. info.
+ *      For IEEE double
+ *          if x >  7.09782712893383973096e+02 then expm1(x) overflow
+ *
+ * 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.
+ */
+
+#include "libm.h"
+
+static const double
+one         = 1.0,
+huge        = 1.0e+300,
+tiny        = 1.0e-300,
+o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
+ln2_hi      = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
+ln2_lo      = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
+invln2      = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
+/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
+Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
+Q2 =  1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
+Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
+Q4 =  4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
+Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
+
+double expm1(double x)
+{
+	double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
+	int32_t k,xsb;
+	uint32_t hx;
+
+	GET_HIGH_WORD(hx, x);
+	xsb = hx&0x80000000;  /* sign bit of x */
+	hx &= 0x7fffffff;     /* high word of |x| */
+
+	/* filter out huge and non-finite argument */
+	if (hx >= 0x4043687A) {  /* if |x|>=56*ln2 */
+		if (hx >= 0x40862E42) {  /* if |x|>=709.78... */
+			if (hx >= 0x7ff00000) {
+				uint32_t low;
+
+				GET_LOW_WORD(low, x);
+				if (((hx&0xfffff)|low) != 0) /* NaN */
+					return x+x;
+				return xsb==0 ? x : -1.0; /* exp(+-inf)={inf,-1} */
+			}
+			if(x > o_threshold)
+				return huge*huge; /* overflow */
+		}
+		if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */
+			/* raise inexact */
+			if(x+tiny<0.0)
+				return tiny-one;  /* return -1 */
+		}
+	}
+
+	/* argument reduction */
+	if (hx > 0x3fd62e42) {  /* if  |x| > 0.5 ln2 */
+		if (hx < 0x3FF0A2B2) {  /* and |x| < 1.5 ln2 */
+			if (xsb == 0) {
+				hi = x - ln2_hi;
+				lo = ln2_lo;
+				k =  1;
+			} else {
+				hi = x + ln2_hi;
+				lo = -ln2_lo;
+				k = -1;
+			}
+		} else {
+			k  = invln2*x + (xsb==0 ? 0.5 : -0.5);
+			t  = k;
+			hi = x - t*ln2_hi;  /* t*ln2_hi is exact here */
+			lo = t*ln2_lo;
+		}
+		STRICT_ASSIGN(double, x, hi - lo);
+		c = (hi-x)-lo;
+	} else if (hx < 0x3c900000) {  /* |x| < 2**-54, return x */
+		/* raise inexact flags when x != 0 */
+		t = huge+x;
+		return x - (t-(huge+x));
+	} else
+		k = 0;
+
+	/* x is now in primary range */
+	hfx = 0.5*x;
+	hxs = x*hfx;
+	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
+	t  = 3.0-r1*hfx;
+	e  = hxs*((r1-t)/(6.0 - x*t));
+	if (k == 0)   /* c is 0 */
+		return x - (x*e-hxs);
+	INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);  /* 2^k */
+	e  = x*(e-c) - c;
+	e -= hxs;
+	if (k == -1)
+		return 0.5*(x-e) - 0.5;
+	if (k == 1) {
+		if (x < -0.25)
+			return -2.0*(e-(x+0.5));
+		return one+2.0*(x-e);
+	}
+	if (k <= -2 || k > 56) {  /* suffice to return exp(x)-1 */
+		y = one - (e-x);
+		if (k == 1024)
+			y = y*2.0*0x1p1023;
+		else
+			y = y*twopk;
+		return y - one;
+	}
+	t = one;
+	if (k < 20) {
+		SET_HIGH_WORD(t, 0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
+		y = t-(e-x);
+		y = y*twopk;
+	} else {
+		SET_HIGH_WORD(t, ((0x3ff-k)<<20));  /* 2^-k */
+		y = x-(e+t);
+		y += one;
+		y = y*twopk;
+	}
+	return y;
+}