mirror of git://git.musl-libc.org/musl
math: exp.c clean up
overflow and underflow was incorrect when the result was not stored. an optimization for the 0.5*ln2 < |x| < 1.5*ln2 domain was removed. did various cleanups around static constants and made the comments consistent with the code.
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121
src/math/exp.c
121
src/math/exp.c
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@ -25,7 +25,7 @@
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* the interval [0,0.34658]:
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* Write
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* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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* We use a special Remes algorithm on [0,0.34658] to generate
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* We use a special Remez algorithm on [0,0.34658] to generate
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* a polynomial of degree 5 to approximate R. The maximum error
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* of this polynomial approximation is bounded by 2**-59. In
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* other words,
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@ -36,15 +36,15 @@
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* | 2.0+P1*z+...+P5*z - R(z) | <= 2
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* | |
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* The computation of exp(r) thus becomes
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* 2*r
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* exp(r) = 1 + -------
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* R - r
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* r*R1(r)
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* 2*r
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* exp(r) = 1 + ----------
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* R(r) - r
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* r*c(r)
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* = 1 + r + ----------- (for better accuracy)
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* 2 - R1(r)
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* 2 - c(r)
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* where
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* 2 4 10
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* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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* 2 4 10
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* c(r) = r - (P1*r + P2*r + ... + P5*r ).
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*
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* 3. Scale back to obtain exp(x):
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* From step 1, we have
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@ -61,27 +61,16 @@
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*
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* Misc. info.
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* For IEEE double
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* if x > 7.09782712893383973096e+02 then exp(x) overflow
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* if x < -7.45133219101941108420e+02 then exp(x) underflow
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*
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* Constants:
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* The hexadecimal values are the intended ones for the following
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* constants. The decimal values may be used, provided that the
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* compiler will convert from decimal to binary accurately enough
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* to produce the hexadecimal values shown.
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* if x > 709.782712893383973096 then exp(x) overflows
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* if x < -745.133219101941108420 then exp(x) underflows
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*/
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#include "libm.h"
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static const double
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halF[2] = {0.5,-0.5,},
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huge = 1.0e+300,
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o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
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u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
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ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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-6.93147180369123816490e-01},/* 0xbfe62e42, 0xfee00000 */
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ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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-1.90821492927058770002e-10},/* 0xbdea39ef, 0x35793c76 */
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half[2] = {0.5,-0.5},
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ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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@ -89,68 +78,56 @@ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
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static const volatile double
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twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */
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double exp(double x)
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{
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double y,hi=0.0,lo=0.0,c,t,twopk;
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int32_t k=0,xsb;
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double hi, lo, c, z;
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int k, sign;
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uint32_t hx;
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GET_HIGH_WORD(hx, x);
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xsb = (hx>>31)&1; /* sign bit of x */
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sign = hx>>31;
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hx &= 0x7fffffff; /* high word of |x| */
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/* filter out non-finite argument */
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if (hx >= 0x40862E42) { /* if |x| >= 709.78... */
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if (hx >= 0x7ff00000) {
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uint32_t lx;
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GET_LOW_WORD(lx,x);
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if (((hx&0xfffff)|lx) != 0) /* NaN */
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return x+x;
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return xsb==0 ? x : 0.0; /* exp(+-inf)={inf,0} */
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/* special cases */
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if (hx >= 0x40862e42) { /* if |x| >= 709.78... */
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if (isnan(x))
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return x;
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if (x > 709.782712893383973096) {
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/* overflow if x!=inf */
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STRICT_ASSIGN(double, x, 0x1p1023 * x);
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return x;
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}
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if (x < -745.13321910194110842) {
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/* underflow if x!=-inf */
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STRICT_ASSIGN(double, x, 0x1p-1000 / -x * 0x1p-1000);
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return x;
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}
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if (x > o_threshold)
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return huge*huge; /* overflow */
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if (x < u_threshold)
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return twom1000*twom1000; /* underflow */
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}
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/* argument reduction */
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if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
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hi = x-ln2HI[xsb];
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lo = ln2LO[xsb];
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k = 1 - xsb - xsb;
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} else {
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k = (int)(invln2*x+halF[xsb]);
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t = k;
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hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
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lo = t*ln2LO[0];
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}
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if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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if (hx < 0x3ff0a2b2) /* if |x| < 1.5 ln2 */
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k = 1 - sign - sign; /* optimization */
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else
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k = (int)(invln2*x + half[sign]);
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hi = x - k*ln2hi; /* k*ln2hi is exact here */
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lo = k*ln2lo;
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STRICT_ASSIGN(double, x, hi - lo);
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} else if(hx < 0x3e300000) { /* |x| < 2**-28 */
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/* raise inexact */
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if (huge+x > 1.0)
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return 1.0+x;
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} else
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} else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
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k = 0;
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hi = x;
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lo = 0;
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} else {
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/* inexact if x!=0 */
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FORCE_EVAL(0x1p1023 + x);
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return 1 + x;
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}
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/* x is now in primary range */
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t = x*x;
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if (k >= -1021)
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INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);
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else
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INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0);
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c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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z = x*x;
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c = x - z*(P1+z*(P2+z*(P3+z*(P4+z*P5))));
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x = 1 + ((x*c/(2-c) - lo) + hi);
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if (k == 0)
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return 1.0 - ((x*c)/(c-2.0) - x);
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y = 1.0-((lo-(x*c)/(2.0-c))-hi);
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if (k < -1021)
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return y*twopk*twom1000;
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if (k == 1024)
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return y*2.0*0x1p1023;
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return y*twopk;
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return x;
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return scalbn(x, k);
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}
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