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-rw-r--r--src/math/exp.c121
1 files changed, 49 insertions, 72 deletions
 diff --git a/src/math/exp.c b/src/math/exp.cindex 29bf9609..5c0edee4 100644--- a/src/math/exp.c+++ b/src/math/exp.c@@ -25,7 +25,7 @@ * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...- * We use a special Remes algorithm on [0,0.34658] to generate+ * We use a special Remez algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words,@@ -36,15 +36,15 @@ * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes- * 2*r- * exp(r) = 1 + -------- * R - r- * r*R1(r)+ * 2*r+ * exp(r) = 1 + ----------+ * R(r) - r+ * r*c(r) * = 1 + r + ----------- (for better accuracy)- * 2 - R1(r)+ * 2 - c(r) * where- * 2 4 10- * R1(r) = r - (P1*r + P2*r + ... + P5*r ).+ * 2 4 10+ * c(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have@@ -61,27 +61,16 @@ * * Misc. info. * For IEEE double- * if x > 7.09782712893383973096e+02 then exp(x) overflow- * if x < -7.45133219101941108420e+02 then exp(x) underflow- *- * 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.+ * if x > 709.782712893383973096 then exp(x) overflows+ * if x < -745.133219101941108420 then exp(x) underflows */ #include "libm.h" static const double-halF[2] = {0.5,-0.5,},-huge = 1.0e+300,-o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */-u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */-ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */- -6.93147180369123816490e-01},/* 0xbfe62e42, 0xfee00000 */-ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */- -1.90821492927058770002e-10},/* 0xbdea39ef, 0x35793c76 */+half[2] = {0.5,-0.5},+ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */+ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */@@ -89,68 +78,56 @@ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ -static const volatile double-twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */- double exp(double x) {- double y,hi=0.0,lo=0.0,c,t,twopk;- int32_t k=0,xsb;+ double hi, lo, c, z;+ int k, sign; uint32_t hx; GET_HIGH_WORD(hx, x);- xsb = (hx>>31)&1; /* sign bit of x */+ sign = hx>>31; hx &= 0x7fffffff; /* high word of |x| */ - /* filter out non-finite argument */- if (hx >= 0x40862E42) { /* if |x| >= 709.78... */- if (hx >= 0x7ff00000) {- uint32_t lx;- - GET_LOW_WORD(lx,x);- if (((hx&0xfffff)|lx) != 0) /* NaN */- return x+x;- return xsb==0 ? x : 0.0; /* exp(+-inf)={inf,0} */+ /* special cases */+ if (hx >= 0x40862e42) { /* if |x| >= 709.78... */+ if (isnan(x))+ return x;+ if (x > 709.782712893383973096) {+ /* overflow if x!=inf */+ STRICT_ASSIGN(double, x, 0x1p1023 * x);+ return x;+ }+ if (x < -745.13321910194110842) {+ /* underflow if x!=-inf */+ STRICT_ASSIGN(double, x, 0x1p-1000 / -x * 0x1p-1000);+ return x; }- if (x > o_threshold)- return huge*huge; /* overflow */- if (x < u_threshold)- return twom1000*twom1000; /* underflow */ } /* argument reduction */- if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */- if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */- hi = x-ln2HI[xsb];- lo = ln2LO[xsb];- k = 1 - xsb - xsb;- } else {- k = (int)(invln2*x+halF[xsb]);- t = k;- hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */- lo = t*ln2LO[0];- }+ if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */+ if (hx < 0x3ff0a2b2) /* if |x| < 1.5 ln2 */+ k = 1 - sign - sign; /* optimization */+ else+ k = (int)(invln2*x + half[sign]);+ hi = x - k*ln2hi; /* k*ln2hi is exact here */+ lo = k*ln2lo; STRICT_ASSIGN(double, x, hi - lo);- } else if(hx < 0x3e300000) { /* |x| < 2**-28 */- /* raise inexact */- if (huge+x > 1.0)- return 1.0+x;- } else+ } else if (hx > 0x3e300000) { /* if |x| > 2**-28 */ k = 0;+ hi = x;+ lo = 0;+ } else {+ /* inexact if x!=0 */+ FORCE_EVAL(0x1p1023 + x);+ return 1 + x;+ } /* x is now in primary range */- t = x*x;- if (k >= -1021)- INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);- else- INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0);- c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));+ z = x*x;+ c = x - z*(P1+z*(P2+z*(P3+z*(P4+z*P5))));+ x = 1 + ((x*c/(2-c) - lo) + hi); if (k == 0)- return 1.0 - ((x*c)/(c-2.0) - x);- y = 1.0-((lo-(x*c)/(2.0-c))-hi);- if (k < -1021)- return y*twopk*twom1000;- if (k == 1024)- return y*2.0*0x1p1023;- return y*twopk;+ return x;+ return scalbn(x, k); }