Fixed missing references to fdlibm functions.
git-svn-id: http://picoc.googlecode.com/svn/trunk@307 21eae674-98b7-11dd-bd71-f92a316d2d60
This commit is contained in:
zik.saleeba
parent
4bc228c439
commit
c1381adae5
325
math_library.c
325
math_library.c
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@ -153,7 +153,19 @@ T[] = {
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7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
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-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
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2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
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};
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},
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Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
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Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
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Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
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Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
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Q5 = -2.01099218183624371326e-07, /* BE8AFDB7 6E09C32D */
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Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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/*
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@ -1451,6 +1463,186 @@ double math_atan(double x)
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}
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/* expm1(x)
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* Returns exp(x)-1, the exponential of x minus 1.
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*
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* Method
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* 1. Argument reduction:
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* Given x, find r and integer k such that
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*
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* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
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*
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* Here a correction term c will be computed to compensate
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* the error in r when rounded to a floating-point number.
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*
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* 2. Approximating expm1(r) by a special rational function on
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* the interval [0,0.34658]:
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* Since
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* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
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* we define R1(r*r) by
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* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
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* That is,
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* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
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* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
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* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
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* We use a special Remes algorithm on [0,0.347] to generate
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* a polynomial of degree 5 in r*r to approximate R1. The
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* maximum error of this polynomial approximation is bounded
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* by 2**-61. In other words,
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* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
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* where Q1 = -1.6666666666666567384E-2,
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* Q2 = 3.9682539681370365873E-4,
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* Q3 = -9.9206344733435987357E-6,
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* Q4 = 2.5051361420808517002E-7,
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* Q5 = -6.2843505682382617102E-9;
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* (where z=r*r, and the values of Q1 to Q5 are listed below)
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* with error bounded by
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* | 5 | -61
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* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
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* | |
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*
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* expm1(r) = exp(r)-1 is then computed by the following
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* specific way which minimize the accumulation rounding error:
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* 2 3
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* r r [ 3 - (R1 + R1*r/2) ]
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* expm1(r) = r + --- + --- * [--------------------]
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* 2 2 [ 6 - r*(3 - R1*r/2) ]
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*
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* To compensate the error in the argument reduction, we use
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* expm1(r+c) = expm1(r) + c + expm1(r)*c
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* ~ expm1(r) + c + r*c
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* Thus c+r*c will be added in as the correction terms for
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* expm1(r+c). Now rearrange the term to avoid optimization
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* screw up:
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* ( 2 2 )
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* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
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* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
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* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
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* ( )
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*
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* = r - E
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* 3. Scale back to obtain expm1(x):
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* From step 1, we have
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* expm1(x) = either 2^k*[expm1(r)+1] - 1
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* = or 2^k*[expm1(r) + (1-2^-k)]
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* 4. Implementation notes:
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* (A). To save one multiplication, we scale the coefficient Qi
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* to Qi*2^i, and replace z by (x^2)/2.
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* (B). To achieve maximum accuracy, we compute expm1(x) by
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* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
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* (ii) if k=0, return r-E
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* (iii) if k=-1, return 0.5*(r-E)-0.5
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* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
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* else return 1.0+2.0*(r-E);
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* (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
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* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
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* (vii) return 2^k(1-((E+2^-k)-r))
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*
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* Special cases:
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* expm1(INF) is INF, expm1(NaN) is NaN;
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* expm1(-INF) is -1, and
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* for finite argument, only expm1(0)=0 is exact.
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*
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* Accuracy:
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* according to an error analysis, the error is always less than
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* 1 ulp (unit in the last place).
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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 expm1(x) overflow
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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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*/
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double math_expm1(double x)
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{
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double y,hi,lo,c,t,e,hxs,hfx,r1;
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int k,xsb;
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unsigned hx;
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hx = __HI(x); /* high word of x */
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xsb = hx&0x80000000; /* sign bit of x */
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if(xsb==0) y=x; else y= -x; /* y = |x| */
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hx &= 0x7fffffff; /* high word of |x| */
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/* filter out huge and non-finite argument */
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if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
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if(hx >= 0x40862E42) { /* if |x|>=709.78... */
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if(hx>=0x7ff00000) {
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if(((hx&0xfffff)|__LO(x))!=0)
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return x+x; /* NaN */
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else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
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}
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if(x > o_threshold) return huge*huge; /* overflow */
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}
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if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
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if(x+tiny<0.0) /* raise inexact */
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return tiny-one; /* return -1 */
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}
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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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if(xsb==0)
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{hi = x - ln2_hi; lo = ln2_lo; k = 1;}
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else
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{hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
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} else {
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k = invln2*x+((xsb==0)?0.5:-0.5);
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t = k;
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hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
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lo = t*ln2_lo;
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}
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x = hi - lo;
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c = (hi-x)-lo;
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}
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else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
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t = huge+x; /* return x with inexact flags when x!=0 */
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return x - (t-(huge+x));
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}
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else k = 0;
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/* x is now in primary range */
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hfx = 0.5*x;
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hxs = x*hfx;
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r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
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t = 3.0-r1*hfx;
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e = hxs*((r1-t)/(6.0 - x*t));
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if(k==0) return x - (x*e-hxs); /* c is 0 */
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else {
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e = (x*(e-c)-c);
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e -= hxs;
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if(k== -1) return 0.5*(x-e)-0.5;
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if(k==1) {
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if(x < -0.25) return -2.0*(e-(x+0.5));
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else return one+2.0*(x-e);
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}
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if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
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y = one-(e-x);
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__HI(y) += (k<<20); /* add k to y's exponent */
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return y-one;
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}
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t = one;
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if(k<20) {
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__HI(t) = 0x3ff00000 - (0x200000>>k); /* t=1-2^-k */
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y = t-(e-x);
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__HI(y) += (k<<20); /* add k to y's exponent */
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} else {
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__HI(t) = ((0x3ff-k)<<20); /* 2^-k */
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y = x-(e+t);
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y += one;
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__HI(y) += (k<<20); /* add k to y's exponent */
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}
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}
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return y;
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}
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/* __ieee754_sinh(x)
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* Method :
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* mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
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@ -1672,6 +1864,137 @@ double math_tanh(double x)
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}
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/* double log1p(double x)
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*
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* Method :
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* 1. Argument Reduction: find k and f such that
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* 1+x = 2^k * (1+f),
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* where sqrt(2)/2 < 1+f < sqrt(2) .
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*
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* Note. If k=0, then f=x is exact. However, if k!=0, then f
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* may not be representable exactly. In that case, a correction
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* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
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* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
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* and add back the correction term c/u.
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* (Note: when x > 2**53, one can simply return log(x))
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*
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* 2. Approximation of log1p(f).
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* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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* = 2s + s*R
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* We use a special Reme algorithm on [0,0.1716] to generate
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* a polynomial of degree 14 to approximate R The maximum error
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* of this polynomial approximation is bounded by 2**-58.45. In
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* other words,
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* 2 4 6 8 10 12 14
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* R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
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* (the values of Lp1 to Lp7 are listed in the program)
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* and
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* | 2 14 | -58.45
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* | Lp1*s +...+Lp7*s - R(z) | <= 2
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* | |
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* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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* In order to guarantee error in log below 1ulp, we compute log
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* by
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* log1p(f) = f - (hfsq - s*(hfsq+R)).
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*
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* 3. Finally, log1p(x) = k*ln2 + log1p(f).
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* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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* Here ln2 is split into two floating point number:
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* ln2_hi + ln2_lo,
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* where n*ln2_hi is always exact for |n| < 2000.
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*
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* Special cases:
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* log1p(x) is NaN with signal if x < -1 (including -INF) ;
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* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
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* log1p(NaN) is that NaN with no signal.
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*
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* Accuracy:
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* according to an error analysis, the error is always less than
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* 1 ulp (unit in the last place).
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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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*
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* Note: Assuming log() return accurate answer, the following
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* algorithm can be used to compute log1p(x) to within a few ULP:
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*
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* u = 1+x;
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* if(u==1.0) return x ; else
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* return log(u)*(x/(u-1.0));
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*
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* See HP-15C Advanced Functions Handbook, p.193.
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*/
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double math_log1p(double x)
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{
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double hfsq,f,c,s,z,R,u;
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int k,hx,hu,ax;
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hx = __HI(x); /* high word of x */
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ax = hx&0x7fffffff;
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k = 1;
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if (hx < 0x3FDA827A) { /* x < 0.41422 */
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if(ax>=0x3ff00000) { /* x <= -1.0 */
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if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
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else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
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}
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if(ax<0x3e200000) { /* |x| < 2**-29 */
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if(two54+x>zero /* raise inexact */
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&&ax<0x3c900000) /* |x| < 2**-54 */
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return x;
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else
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return x - x*x*0.5;
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}
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if(hx>0||hx<=((int)0xbfd2bec3)) {
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k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
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}
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if (hx >= 0x7ff00000) return x+x;
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if(k!=0) {
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if(hx<0x43400000) {
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u = 1.0+x;
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hu = __HI(u); /* high word of u */
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k = (hu>>20)-1023;
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c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
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c /= u;
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} else {
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u = x;
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hu = __HI(u); /* high word of u */
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k = (hu>>20)-1023;
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c = 0;
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}
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hu &= 0x000fffff;
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if(hu<0x6a09e) {
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__HI(u) = hu|0x3ff00000; /* normalize u */
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} else {
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k += 1;
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__HI(u) = hu|0x3fe00000; /* normalize u/2 */
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hu = (0x00100000-hu)>>2;
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}
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f = u-1.0;
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}
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hfsq=0.5*f*f;
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if(hu==0) { /* |f| < 2**-20 */
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if(f==zero) {
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if(k==0) return zero;
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else {c += k*ln2_lo; return k*ln2_hi+c;}
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}
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R = hfsq*(1.0-0.66666666666666666*f);
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if(k==0) return f-R; else
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return k*ln2_hi-((R-(k*ln2_lo+c))-f);
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}
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s = f/(2.0+f);
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z = s*s;
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R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
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if(k==0) return f-(hfsq-s*(hfsq+R)); else
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return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
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}
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/* asinh(x)
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* Method :
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* Based on
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