s_expm1.c 7.2 KB

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  1. /*
  2. * ====================================================
  3. * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  4. *
  5. * Developed at SunPro, a Sun Microsystems, Inc. business.
  6. * Permission to use, copy, modify, and distribute this
  7. * software is freely granted, provided that this notice
  8. * is preserved.
  9. * ====================================================
  10. */
  11. /* expm1(x)
  12. * Returns exp(x)-1, the exponential of x minus 1.
  13. *
  14. * Method
  15. * 1. Argument reduction:
  16. * Given x, find r and integer k such that
  17. *
  18. * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
  19. *
  20. * Here a correction term c will be computed to compensate
  21. * the error in r when rounded to a floating-point number.
  22. *
  23. * 2. Approximating expm1(r) by a special rational function on
  24. * the interval [0,0.34658]:
  25. * Since
  26. * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
  27. * we define R1(r*r) by
  28. * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
  29. * That is,
  30. * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
  31. * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
  32. * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
  33. * We use a special Reme algorithm on [0,0.347] to generate
  34. * a polynomial of degree 5 in r*r to approximate R1. The
  35. * maximum error of this polynomial approximation is bounded
  36. * by 2**-61. In other words,
  37. * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
  38. * where Q1 = -1.6666666666666567384E-2,
  39. * Q2 = 3.9682539681370365873E-4,
  40. * Q3 = -9.9206344733435987357E-6,
  41. * Q4 = 2.5051361420808517002E-7,
  42. * Q5 = -6.2843505682382617102E-9;
  43. * (where z=r*r, and the values of Q1 to Q5 are listed below)
  44. * with error bounded by
  45. * | 5 | -61
  46. * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
  47. * | |
  48. *
  49. * expm1(r) = exp(r)-1 is then computed by the following
  50. * specific way which minimize the accumulation rounding error:
  51. * 2 3
  52. * r r [ 3 - (R1 + R1*r/2) ]
  53. * expm1(r) = r + --- + --- * [--------------------]
  54. * 2 2 [ 6 - r*(3 - R1*r/2) ]
  55. *
  56. * To compensate the error in the argument reduction, we use
  57. * expm1(r+c) = expm1(r) + c + expm1(r)*c
  58. * ~ expm1(r) + c + r*c
  59. * Thus c+r*c will be added in as the correction terms for
  60. * expm1(r+c). Now rearrange the term to avoid optimization
  61. * screw up:
  62. * ( 2 2 )
  63. * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
  64. * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
  65. * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
  66. * ( )
  67. *
  68. * = r - E
  69. * 3. Scale back to obtain expm1(x):
  70. * From step 1, we have
  71. * expm1(x) = either 2^k*[expm1(r)+1] - 1
  72. * = or 2^k*[expm1(r) + (1-2^-k)]
  73. * 4. Implementation notes:
  74. * (A). To save one multiplication, we scale the coefficient Qi
  75. * to Qi*2^i, and replace z by (x^2)/2.
  76. * (B). To achieve maximum accuracy, we compute expm1(x) by
  77. * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
  78. * (ii) if k=0, return r-E
  79. * (iii) if k=-1, return 0.5*(r-E)-0.5
  80. * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
  81. * else return 1.0+2.0*(r-E);
  82. * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
  83. * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
  84. * (vii) return 2^k(1-((E+2^-k)-r))
  85. *
  86. * Special cases:
  87. * expm1(INF) is INF, expm1(NaN) is NaN;
  88. * expm1(-INF) is -1, and
  89. * for finite argument, only expm1(0)=0 is exact.
  90. *
  91. * Accuracy:
  92. * according to an error analysis, the error is always less than
  93. * 1 ulp (unit in the last place).
  94. *
  95. * Misc. info.
  96. * For IEEE double
  97. * if x > 7.09782712893383973096e+02 then expm1(x) overflow
  98. *
  99. * Constants:
  100. * The hexadecimal values are the intended ones for the following
  101. * constants. The decimal values may be used, provided that the
  102. * compiler will convert from decimal to binary accurately enough
  103. * to produce the hexadecimal values shown.
  104. */
  105. #include "math.h"
  106. #include "math_private.h"
  107. static const double
  108. one = 1.0,
  109. huge = 1.0e+300,
  110. tiny = 1.0e-300,
  111. o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
  112. ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
  113. ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
  114. invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
  115. /* scaled coefficients related to expm1 */
  116. Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
  117. Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
  118. Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
  119. Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
  120. Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
  121. double expm1(double x)
  122. {
  123. double y,hi,lo,c=0.0,t,e,hxs,hfx,r1;
  124. int32_t k,xsb;
  125. u_int32_t hx;
  126. GET_HIGH_WORD(hx,x);
  127. xsb = hx&0x80000000; /* sign bit of x */
  128. if(xsb==0) y=x; else y= -x; /* y = |x| */
  129. hx &= 0x7fffffff; /* high word of |x| */
  130. /* filter out huge and non-finite argument */
  131. if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
  132. if(hx >= 0x40862E42) { /* if |x|>=709.78... */
  133. if(hx>=0x7ff00000) {
  134. u_int32_t low;
  135. GET_LOW_WORD(low,x);
  136. if(((hx&0xfffff)|low)!=0)
  137. return x+x; /* NaN */
  138. else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
  139. }
  140. if(x > o_threshold) return huge*huge; /* overflow */
  141. }
  142. if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
  143. if(x+tiny<0.0) /* raise inexact */
  144. return tiny-one; /* return -1 */
  145. }
  146. }
  147. /* argument reduction */
  148. if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
  149. if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
  150. if(xsb==0)
  151. {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
  152. else
  153. {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
  154. } else {
  155. k = (int32_t)(invln2*x+((xsb==0)?0.5:-0.5));
  156. t = k;
  157. hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
  158. lo = t*ln2_lo;
  159. }
  160. x = hi - lo;
  161. c = (hi-x)-lo;
  162. }
  163. else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
  164. t = huge+x; /* return x with inexact flags when x!=0 */
  165. return x - (t-(huge+x));
  166. }
  167. else k = 0;
  168. /* x is now in primary range */
  169. hfx = 0.5*x;
  170. hxs = x*hfx;
  171. r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
  172. t = 3.0-r1*hfx;
  173. e = hxs*((r1-t)/(6.0 - x*t));
  174. if(k==0) return x - (x*e-hxs); /* c is 0 */
  175. else {
  176. e = (x*(e-c)-c);
  177. e -= hxs;
  178. if(k== -1) return 0.5*(x-e)-0.5;
  179. if(k==1) {
  180. if(x < -0.25) return -2.0*(e-(x+0.5));
  181. else return one+2.0*(x-e);
  182. }
  183. if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
  184. u_int32_t high;
  185. y = one-(e-x);
  186. GET_HIGH_WORD(high,y);
  187. SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
  188. return y-one;
  189. }
  190. t = one;
  191. if(k<20) {
  192. u_int32_t high;
  193. SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
  194. y = t-(e-x);
  195. GET_HIGH_WORD(high,y);
  196. SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
  197. } else {
  198. u_int32_t high;
  199. SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
  200. y = x-(e+t);
  201. y += one;
  202. GET_HIGH_WORD(high,y);
  203. SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
  204. }
  205. }
  206. return y;
  207. }
  208. libm_hidden_def(expm1)