+++ /dev/null
-/* @(#)s_expm1.c 5.1 93/09/24 */\r
-/*\r
- * ====================================================\r
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\r
- *\r
- * Developed at SunPro, a Sun Microsystems, Inc. business.\r
- * Permission to use, copy, modify, and distribute this\r
- * software is freely granted, provided that this notice\r
- * is preserved.\r
- * ====================================================\r
- */\r
-#include <LibConfig.h>\r
-#include <sys/EfiCdefs.h>\r
-#if defined(LIBM_SCCS) && !defined(lint)\r
-__RCSID("$NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $");\r
-#endif\r
-\r
-#if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */\r
- // C4756: overflow in constant arithmetic\r
- #pragma warning ( disable : 4756 )\r
-#endif\r
-\r
-/* expm1(x)\r
- * Returns exp(x)-1, the exponential of x minus 1.\r
- *\r
- * Method\r
- * 1. Argument reduction:\r
- * Given x, find r and integer k such that\r
- *\r
- * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658\r
- *\r
- * Here a correction term c will be computed to compensate\r
- * the error in r when rounded to a floating-point number.\r
- *\r
- * 2. Approximating expm1(r) by a special rational function on\r
- * the interval [0,0.34658]:\r
- * Since\r
- * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...\r
- * we define R1(r*r) by\r
- * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)\r
- * That is,\r
- * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)\r
- * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))\r
- * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...\r
- * We use a special Reme algorithm on [0,0.347] to generate\r
- * a polynomial of degree 5 in r*r to approximate R1. The\r
- * maximum error of this polynomial approximation is bounded\r
- * by 2**-61. In other words,\r
- * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5\r
- * where Q1 = -1.6666666666666567384E-2,\r
- * Q2 = 3.9682539681370365873E-4,\r
- * Q3 = -9.9206344733435987357E-6,\r
- * Q4 = 2.5051361420808517002E-7,\r
- * Q5 = -6.2843505682382617102E-9;\r
- * (where z=r*r, and the values of Q1 to Q5 are listed below)\r
- * with error bounded by\r
- * | 5 | -61\r
- * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2\r
- * | |\r
- *\r
- * expm1(r) = exp(r)-1 is then computed by the following\r
- * specific way which minimize the accumulation rounding error:\r
- * 2 3\r
- * r r [ 3 - (R1 + R1*r/2) ]\r
- * expm1(r) = r + --- + --- * [--------------------]\r
- * 2 2 [ 6 - r*(3 - R1*r/2) ]\r
- *\r
- * To compensate the error in the argument reduction, we use\r
- * expm1(r+c) = expm1(r) + c + expm1(r)*c\r
- * ~ expm1(r) + c + r*c\r
- * Thus c+r*c will be added in as the correction terms for\r
- * expm1(r+c). Now rearrange the term to avoid optimization\r
- * screw up:\r
- * ( 2 2 )\r
- * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )\r
- * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )\r
- * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )\r
- * ( )\r
- *\r
- * = r - E\r
- * 3. Scale back to obtain expm1(x):\r
- * From step 1, we have\r
- * expm1(x) = either 2^k*[expm1(r)+1] - 1\r
- * = or 2^k*[expm1(r) + (1-2^-k)]\r
- * 4. Implementation notes:\r
- * (A). To save one multiplication, we scale the coefficient Qi\r
- * to Qi*2^i, and replace z by (x^2)/2.\r
- * (B). To achieve maximum accuracy, we compute expm1(x) by\r
- * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)\r
- * (ii) if k=0, return r-E\r
- * (iii) if k=-1, return 0.5*(r-E)-0.5\r
- * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)\r
- * else return 1.0+2.0*(r-E);\r
- * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)\r
- * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else\r
- * (vii) return 2^k(1-((E+2^-k)-r))\r
- *\r
- * Special cases:\r
- * expm1(INF) is INF, expm1(NaN) is NaN;\r
- * expm1(-INF) is -1, and\r
- * for finite argument, only expm1(0)=0 is exact.\r
- *\r
- * Accuracy:\r
- * according to an error analysis, the error is always less than\r
- * 1 ulp (unit in the last place).\r
- *\r
- * Misc. info.\r
- * For IEEE double\r
- * if x > 7.09782712893383973096e+02 then expm1(x) overflow\r
- *\r
- * Constants:\r
- * The hexadecimal values are the intended ones for the following\r
- * constants. The decimal values may be used, provided that the\r
- * compiler will convert from decimal to binary accurately enough\r
- * to produce the hexadecimal values shown.\r
- */\r
-\r
-#include "math.h"\r
-#include "math_private.h"\r
-\r
-static const double\r
-one = 1.0,\r
-huge = 1.0e+300,\r
-tiny = 1.0e-300,\r
-o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */\r
-ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */\r
-ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */\r
-invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */\r
- /* scaled coefficients related to expm1 */\r
-Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */\r
-Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */\r
-Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */\r
-Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */\r
-Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */\r
-\r
-double\r
-expm1(double x)\r
-{\r
- double y,hi,lo,c,t,e,hxs,hfx,r1;\r
- int32_t k,xsb;\r
- u_int32_t hx;\r
-\r
- c = 0;\r
- GET_HIGH_WORD(hx,x);\r
- xsb = hx&0x80000000; /* sign bit of x */\r
- if(xsb==0) y=x; else y= -x; /* y = |x| */\r
- hx &= 0x7fffffff; /* high word of |x| */\r
-\r
- /* filter out huge and non-finite argument */\r
- if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */\r
- if(hx >= 0x40862E42) { /* if |x|>=709.78... */\r
- if(hx>=0x7ff00000) {\r
- u_int32_t low;\r
- GET_LOW_WORD(low,x);\r
- if(((hx&0xfffff)|low)!=0)\r
- return x+x; /* NaN */\r
- else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */\r
- }\r
- if(x > o_threshold) return huge*huge; /* overflow */\r
- }\r
- if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */\r
- if(x+tiny<0.0) /* raise inexact */\r
- return tiny-one; /* return -1 */\r
- }\r
- }\r
-\r
- /* argument reduction */\r
- if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */\r
- if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */\r
- if(xsb==0)\r
- {hi = x - ln2_hi; lo = ln2_lo; k = 1;}\r
- else\r
- {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}\r
- } else {\r
- k = (int32_t)(invln2*x+((xsb==0)?0.5:-0.5));\r
- t = k;\r
- hi = x - t*ln2_hi; /* t*ln2_hi is exact here */\r
- lo = t*ln2_lo;\r
- }\r
- x = hi - lo;\r
- c = (hi-x)-lo;\r
- }\r
- else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */\r
- t = huge+x; /* return x with inexact flags when x!=0 */\r
- return x - (t-(huge+x));\r
- }\r
- else k = 0;\r
-\r
- /* x is now in primary range */\r
- hfx = 0.5*x;\r
- hxs = x*hfx;\r
- r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));\r
- t = 3.0-r1*hfx;\r
- e = hxs*((r1-t)/(6.0 - x*t));\r
- if(k==0) return x - (x*e-hxs); /* c is 0 */\r
- else {\r
- e = (x*(e-c)-c);\r
- e -= hxs;\r
- if(k== -1) return 0.5*(x-e)-0.5;\r
- if(k==1) {\r
- if(x < -0.25) return -2.0*(e-(x+0.5));\r
- else return one+2.0*(x-e);\r
- }\r
- if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */\r
- u_int32_t high;\r
- y = one-(e-x);\r
- GET_HIGH_WORD(high,y);\r
- SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */\r
- return y-one;\r
- }\r
- t = one;\r
- if(k<20) {\r
- u_int32_t high;\r
- SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */\r
- y = t-(e-x);\r
- GET_HIGH_WORD(high,y);\r
- SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */\r
- } else {\r
- u_int32_t high;\r
- SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */\r
- y = x-(e+t);\r
- y += one;\r
- GET_HIGH_WORD(high,y);\r
- SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */\r
- }\r
- }\r
- return y;\r
-}\r
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