more accurate Log, Exp, Pow.

move test.go to alll_test.go.

R=r
DELTA=1024  (521 added, 425 deleted, 78 changed)
OCL=19687
CL=19695

6 files changed, 329 insertions, 136 deletions

diff --git a/src/lib/math/Makefile b/src/lib/math/Makefile
index b24dbca7a..672d17fac 100644
--- a/src/lib/math/Makefile
+++ b/src/lib/math/Makefile
@@ -33,6 +33,7 @@ coverage: packages
 
 O1=\
 	atan.$O\
+	exp.$O\
 	fabs.$O\
 	floor.$O\
 	fmod.$O\
@@ -46,32 +47,25 @@ O1=\
 O2=\
 	asin.$O\
 	atan2.$O\
-	exp.$O\
-
-O3=\
 	pow.$O\
 	sinh.$O\
 
-O4=\
+O3=\
 	tanh.$O\
 
-math.a: a1 a2 a3 a4
+math.a: a1 a2 a3
 
 a1:	$(O1)
-	$(AR) grc math.a atan.$O fabs.$O floor.$O fmod.$O hypot.$O log.$O pow10.$O sin.$O sqrt.$O tan.$O
+	$(AR) grc math.a atan.$O exp.$O fabs.$O floor.$O fmod.$O hypot.$O log.$O pow10.$O sin.$O sqrt.$O tan.$O
 	rm -f $(O1)
 
 a2:	$(O2)
-	$(AR) grc math.a asin.$O atan2.$O exp.$O
+	$(AR) grc math.a asin.$O atan2.$O pow.$O sinh.$O
 	rm -f $(O2)
 
 a3:	$(O3)
-	$(AR) grc math.a pow.$O sinh.$O
-	rm -f $(O3)
-
-a4:	$(O4)
 	$(AR) grc math.a tanh.$O
-	rm -f $(O4)
+	rm -f $(O3)
 
 newpkg: clean
 	$(AR) grc math.a
@@ -79,7 +73,6 @@ newpkg: clean
 $(O1): newpkg
 $(O2): a1
 $(O3): a2
-$(O4): a3
 
 nuke: clean
 	rm -f $(GOROOT)/pkg/math.a
diff --git a/src/lib/math/test.go b/src/lib/math/all_test.go
index b4eb8bb0e..8fa334c35 100644
--- a/src/lib/math/test.go
+++ b/src/lib/math/all_test.go
@@ -50,7 +50,7 @@ var atan = []float64 {
 var exp = []float64 {
 	  1.4533071302642137e+02,
 	  2.2958822575694450e+03,
-	  7.5814542574851664e-01,
+	  7.5814542574851666e-01,
 	  6.6668778421791010e-03,
 	  1.5310493273896035e+04,
 	  1.8659907517999329e+01,
@@ -156,13 +156,12 @@ var tanh = []float64 {
 	 -9.9999994291374019e-01,
 }
 
-func Close(a,b float64) bool {
+func Tolerance(a,b,e float64) bool {
 	d := a-b;
 	if d < 0 {
 		d = -d;
 	}
 
-	e := float64(1e-14);
 	if a != 0 {
 		e = e*a;
 		if e < 0 {
@@ -171,10 +170,16 @@ func Close(a,b float64) bool {
 	}
 	return d < e;
 }
+func Close(a,b float64) bool {
+	return Tolerance(a, b, 1e-14);
+}
+func VeryClose(a,b float64) bool {
+	return Tolerance(a, b, 4e-16);
+}
 
 export func TestAsin(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := math.Asin(vf[i]/10); !Close(asin[i], f) {
+		if f := math.Asin(vf[i]/10); !VeryClose(asin[i], f) {
 			t.Errorf("math.Asin(%g) = %g, want %g\n", vf[i]/10, f, asin[i]);
 		}
 	}
@@ -182,7 +187,7 @@ export func TestAsin(t *testing.T) {
 
 export func TestAtan(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := math.Atan(vf[i]); !Close(atan[i], f) {
+		if f := math.Atan(vf[i]); !VeryClose(atan[i], f) {
 			t.Errorf("math.Atan(%g) = %g, want %g\n", vf[i], f, atan[i]);
 		}
 	}
@@ -190,7 +195,7 @@ export func TestAtan(t *testing.T) {
 
 export func TestExp(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := math.Exp(vf[i]); !Close(exp[i], f) {
+		if f := math.Exp(vf[i]); !VeryClose(exp[i], f) {
 			t.Errorf("math.Exp(%g) = %g, want %g\n", vf[i], f, exp[i]);
 		}
 	}
@@ -198,7 +203,7 @@ export func TestExp(t *testing.T) {
 
 export func TestFloor(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := math.Floor(vf[i]); !Close(floor[i], f) {
+		if f := math.Floor(vf[i]); floor[i] != f {
 			t.Errorf("math.Floor(%g) = %g, want %g\n", vf[i], f, floor[i]);
 		}
 	}
@@ -207,10 +212,14 @@ export func TestFloor(t *testing.T) {
 export func TestLog(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
 		a := math.Fabs(vf[i]);
-		if f := math.Log(a); !Close(log[i], f) {
-			t.Errorf("math.Log(%g) = %g, want %g\n", a, f, floor[i]);
+		if f := math.Log(a); log[i] != f {
+			t.Errorf("math.Log(%g) = %g, want %g\n", a, f, log[i]);
 		}
 	}
+	const Ln10 = 2.30258509299404568401799145468436421;
+	if f := math.Log(10); f != Ln10 {
+		t.Errorf("math.Log(%g) = %g, want %g\n", 10, f, Ln10);
+	}
 }
 
 export func TestPow(t *testing.T) {
@@ -231,7 +240,7 @@ export func TestSin(t *testing.T) {
 
 export func TestSinh(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := math.Sinh(vf[i]); !Close(sinh[i], f) {
+		if f := math.Sinh(vf[i]); !VeryClose(sinh[i], f) {
 			t.Errorf("math.Sinh(%g) = %g, want %g\n", vf[i], f, sinh[i]);
 		}
 	}
@@ -240,7 +249,7 @@ export func TestSinh(t *testing.T) {
 export func TestSqrt(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
 		a := math.Fabs(vf[i]);
-		if f := math.Sqrt(a); !Close(sqrt[i], f) {
+		if f := math.Sqrt(a); !VeryClose(sqrt[i], f) {
 			t.Errorf("math.Sqrt(%g) = %g, want %g\n", a, f, floor[i]);
 		}
 	}
@@ -256,7 +265,7 @@ export func TestTan(t *testing.T) {
 
 export func TestTanh(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := math.Tanh(vf[i]); !Close(tanh[i], f) {
+		if f := math.Tanh(vf[i]); !VeryClose(tanh[i], f) {
 			t.Errorf("math.Tanh(%g) = %g, want %g\n", vf[i], f, tanh[i]);
 		}
 	}
@@ -265,9 +274,8 @@ export func TestTanh(t *testing.T) {
 export func TestHypot(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
 		a := math.Fabs(tanh[i]*math.Sqrt(2));
-		if f := math.Hypot(tanh[i], tanh[i]); !Close(a, f) {
+		if f := math.Hypot(tanh[i], tanh[i]); !VeryClose(a, f) {
 			t.Errorf("math.Hypot(%g, %g) = %g, want %g\n", tanh[i], tanh[i], f, a);
 		}
 	}
 }
-
diff --git a/src/lib/math/exp.go b/src/lib/math/exp.go
index 9bc26d2b6..e1402f02a 100644
--- a/src/lib/math/exp.go
+++ b/src/lib/math/exp.go
@@ -6,42 +6,132 @@ package math
 
 import "math"
 
-/*
- *	exp returns the exponential func of its
- *	floating-point argument.
- *
- *	The coefficients are #1069 from Hart and Cheney. (22.35D)
- */
-
-const
-(
-	p0	= .2080384346694663001443843411e7;
-	p1	= .3028697169744036299076048876e5;
-	p2	= .6061485330061080841615584556e2;
-	q0	= .6002720360238832528230907598e7;
-	q1	= .3277251518082914423057964422e6;
-	q2	= .1749287689093076403844945335e4;
-	log2e	= .14426950408889634073599247e1;
-	sqrt2	= .14142135623730950488016887e1;
-	maxf	= 10000;
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
+// and came with this notice.  The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+//
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// exp(x)
+// Returns the exponential of x.
+//
+// Method
+//   1. Argument reduction:
+//      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
+//      Given x, find r and integer k such that
+//
+//               x = k*ln2 + r,  |r| <= 0.5*ln2.
+//
+//      Here r will be represented as r = hi-lo for better
+//      accuracy.
+//
+//   2. Approximation of exp(r) by a special rational function on
+//      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
+//      a polynomial of degree 5 to approximate R. The maximum error
+//      of this polynomial approximation is bounded by 2**-59. In
+//      other words,
+//          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
+//      (where z=r*r, and the values of P1 to P5 are listed below)
+//      and
+//          |                  5          |     -59
+//          | 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)
+//                     = 1 + r + ----------- (for better accuracy)
+//                                2 - R1(r)
+//      where
+//                               2       4             10
+//              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
+//
+//   3. Scale back to obtain exp(x):
+//      From step 1, we have
+//         exp(x) = 2^k * exp(r)
+//
+// Special cases:
+//      exp(INF) is INF, exp(NaN) is NaN;
+//      exp(-INF) is 0, and
+//      for finite argument, only exp(0)=1 is exact.
+//
+// Accuracy:
+//      according to an error analysis, the error is always less than
+//      1 ulp (unit in the last place).
+//
+// 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.
+
+export const (
+	Ln2				= 0.693147180559945309417232121458176568;
+	HalfLn2			= 0.346573590279972654708616060729088284;
+
+	Ln2Hi	= 6.93147180369123816490e-01;
+	Ln2Lo	= 1.90821492927058770002e-10;
+	Log2e	= 1.44269504088896338700e+00;
+
+	P1   =  1.66666666666666019037e-01; /* 0x3FC55555; 0x5555553E */
+	P2   = -2.77777777770155933842e-03; /* 0xBF66C16C; 0x16BEBD93 */
+	P3   =  6.61375632143793436117e-05; /* 0x3F11566A; 0xAF25DE2C */
+	P4   = -1.65339022054652515390e-06; /* 0xBEBBBD41; 0xC5D26BF1 */
+	P5   =  4.13813679705723846039e-08; /* 0x3E663769; 0x72BEA4D0 */
+
+	Overflow	= 7.09782712893383973096e+02;
+	Underflow	= -7.45133219101941108420e+02;
+	NearZero	= 1.0/(1<<28);		// 2^-28
 )
 
-export func Exp(arg float64) float64 {
-	if arg == 0. {
-		return 1;
-	}
-	if arg < -maxf {
+export func Exp(x float64) float64 {
+	// special cases
+	switch {
+	case sys.isNaN(x) || sys.isInf(x, 1):
+		return x;
+	case sys.isInf(x, -1):
+		return 0;
+	case x > Overflow:
+		return sys.Inf(1);
+	case x < Underflow:
 		return 0;
+	case -NearZero < x && x < NearZero:
+		return 1;
 	}
-	if arg > maxf {
-		return sys.Inf(1)
+
+	// reduce; computed as r = hi - lo for extra precision.
+	var k int;
+	switch {
+	case x < 0:
+		k = int(Log2e*x - 0.5);
+	case x > 0:
+		k = int(Log2e*x + 0.5);
 	}
+	hi := x - float64(k)*Ln2Hi;
+	lo := float64(k)*Ln2Lo;
+	r := hi - lo;
 
-	x := arg*log2e;
-	ent := int(Floor(x));
-	fract := (x-float64(ent)) - 0.5;
-	xsq := fract*fract;
-	temp1 := ((p2*xsq+p1)*xsq+p0)*fract;
-	temp2 := ((xsq+q2)*xsq+q1)*xsq + q0;
-	return sys.ldexp(sqrt2*(temp2+temp1)/(temp2-temp1), ent);
+	// compute
+	t := r * r;
+	c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
+	y := 1 - ((lo - (r*c)/(2-c)) - hi);
+	// TODO(rsc): make sure sys.ldexp can handle boundary k
+	return sys.ldexp(y, k);
 }
diff --git a/src/lib/math/log.go b/src/lib/math/log.go
index e51c72980..c0cfebf8b 100644
--- a/src/lib/math/log.go
+++ b/src/lib/math/log.go
@@ -4,56 +4,128 @@
 
 package math
 
-/*
- *	Log returns the natural logarithm of its floating
- *	point argument.
- *
- *	The coefficients are #2705 from Hart & Cheney. (19.38D)
- *
- *	It calls frexp.
- */
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
+// and came with this notice.  The go code is a simpler
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_log(x)
+// Return the logrithm of x
+//
+// Method :
+//   1. Argument Reduction: find k and f such that
+//			x = 2^k * (1+f),
+//	   where  sqrt(2)/2 < 1+f < sqrt(2) .
+//
+//   2. Approximation of log(1+f).
+//	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+//		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+//	     	 = 2s + s*R
+//      We use a special Reme algorithm on [0,0.1716] to generate
+// 	a polynomial of degree 14 to approximate R The maximum error
+//	of this polynomial approximation is bounded by 2**-58.45. In
+//	other words,
+//		        2      4      6      8      10      12      14
+//	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
+//  	(the values of Lg1 to Lg7 are listed in the program)
+//	and
+//	    |      2          14          |     -58.45
+//	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
+//	    |                             |
+//	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+//	In order to guarantee error in log below 1ulp, we compute log
+//	by
+//		log(1+f) = f - s*(f - R)	(if f is not too large)
+//		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
+//
+//	3. Finally,  log(x) = k*ln2 + log(1+f).
+//			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
+//	   Here ln2 is split into two floating point number:
+//			ln2_hi + ln2_lo,
+//	   where n*ln2_hi is always exact for |n| < 2000.
+//
+// Special cases:
+//	log(x) is NaN with signal if x < 0 (including -INF) ;
+//	log(+INF) is +INF; log(0) is -INF with signal;
+//	log(NaN) is that NaN with no signal.
+//
+// Accuracy:
+//	according to an error analysis, the error is always less than
+//	1 ulp (unit in the last place).
+//
+// 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.
 
-const
-(
-	log2	=   .693147180559945309e0;
-	ln10u1	=   .4342944819032518276511;
-	sqrto2	=   .707106781186547524e0;
-	p0	=  -.240139179559210510e2;
-	p1	=   .309572928215376501e2;
-	p2	=  -.963769093377840513e1;
-	p3	=   .421087371217979714e0;
-	q0	=  -.120069589779605255e2;
-	q1	=   .194809660700889731e2;
-	q2	=  -.891110902798312337e1;
+const (
+	Ln2Hi = 6.93147180369123816490e-01;	/* 3fe62e42 fee00000 */
+	Ln2Lo = 1.90821492927058770002e-10;	/* 3dea39ef 35793c76 */
+	Lg1 = 6.666666666666735130e-01;  /* 3FE55555 55555593 */
+	Lg2 = 3.999999999940941908e-01;  /* 3FD99999 9997FA04 */
+	Lg3 = 2.857142874366239149e-01;  /* 3FD24924 94229359 */
+	Lg4 = 2.222219843214978396e-01;  /* 3FCC71C5 1D8E78AF */
+	Lg5 = 1.818357216161805012e-01;  /* 3FC74664 96CB03DE */
+	Lg6 = 1.531383769920937332e-01;  /* 3FC39A09 D078C69F */
+	Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
+
+	Two54 = 1<<54;				// 2^54
+	TwoM20 = 1.0/(1<<20);		// 2^-20
+	TwoM1022 = 2.2250738585072014e-308;	// 2^-1022
+	Sqrt2 = 1.41421356237309504880168872420969808;
 )
 
-export func Log(arg float64) float64 {
-	if arg <= 0 {
+export func Log(x float64) float64 {
+	// special cases
+	switch {
+	case sys.isNaN(x) || sys.isInf(x, 1):
+		return x;
+	case x < 0:
 		return sys.NaN();
+	case x == 0:
+		return sys.Inf(-1);
 	}
 
-	x, exp := sys.frexp(arg);
-	for x < 0.5 {
-		x = x*2;
-		exp = exp-1;
-	}
-	if x < sqrto2 {
-		x = x*2;
-		exp = exp-1;
+	// reduce
+	f1, ki := sys.frexp(x);
+	if f1 < Sqrt2/2 {
+		f1 *= 2;
+		ki--;
 	}
+	f := f1 - 1;
+	k := float64(ki);
 
-	z := (x-1) / (x+1);
-	zsq := z*z;
-
-	temp := ((p3*zsq + p2)*zsq + p1)*zsq + p0;
-	temp = temp/(((zsq + q2)*zsq + q1)*zsq + q0);
-	temp = temp*z + float64(exp)*log2;
-	return temp;
+	// compute
+	s := f/(2+f);
+	s2 := s*s;
+	s4 := s2*s2;
+	t1 := s2*(Lg1 + s4*(Lg3 + s4*(Lg5 + s4*Lg7)));
+	t2 := s4*(Lg2 + s4*(Lg4 + s4*Lg6));
+	R :=  t1 + t2;
+	hfsq := 0.5*f*f;
+	return k*Ln2Hi - ((hfsq-(s*(hfsq+R)+k*Ln2Lo)) - f);
 }
 
+const
+(
+	ln10u1	= .4342944819032518276511;
+)
+
 export func Log10(arg float64) float64 {
 	if arg <= 0 {
 		return sys.NaN();
 	}
 	return Log(arg) * ln10u1;
 }
+
+
diff --git a/src/lib/math/pow.go b/src/lib/math/pow.go
index 9e63db74b..bdecf1329 100644
--- a/src/lib/math/pow.go
+++ b/src/lib/math/pow.go
@@ -6,56 +6,82 @@ package math
 
 import "math"
 
-/*
-	arg1 ^ arg2 (exponentiation)
- */
+// x^y: exponentation
+export func Pow(x, y float64) float64 {
+	// TODO: x or y NaN, ±Inf, maybe ±0.
+	switch {
+	case y == 0:
+		return 1;
+	case y == 1:
+		return x;
+	case x == 0 && y > 0:
+		return 0;
+	case x == 0 && y < 0:
+		return sys.Inf(1);
+	case y == 0.5:
+		return Sqrt(x);
+	case y == -0.5:
+		return 1 / Sqrt(x);
+	}
 
-export func Pow(arg1,arg2 float64) float64 {
-	if arg2 < 0 {
-		return 1/Pow(arg1, -arg2);
+	absy := y;
+	flip := false;
+	if absy < 0 {
+		absy = -absy;
+		flip = true;
+	}
+	yi, yf := sys.modf(absy);
+	if yf != 0 && x < 0 {
+		return sys.NaN();
+	}
+	if yi >= 1<<63 {
+		return Exp(y * Log(x));
 	}
-	if arg1 <= 0 {
-		if(arg1 == 0) {
-			if arg2 <= 0 {
-				return sys.NaN();
-			}
-			return 0;
-		}
 
-		temp := Floor(arg2);
-		if temp != arg2 {
-			panic(sys.NaN());
-		}
+	ans := float64(1);
 
-		l := int32(temp);
-		if l&1 != 0 {
-			return -Pow(-arg1, arg2);
+	// ans *= x^yf
+	if yf != 0 {
+		if yf > 0.5 {
+			yf--;
+			yi++;
 		}
-		return Pow(-arg1, arg2);
+		ans = Exp(yf * Log(x));
 	}
 
-	temp := Floor(arg2);
-	if temp != arg2 {
-		if arg2-temp == .5 {
-			if temp == 0 {
-				return Sqrt(arg1);
+	// ans *= x^yi
+	// by multiplying in successive squarings
+	// of x according to bits of yi.
+	// accumulate powers of two into exp.
+	// will still have to do ans *= 2^exp later.
+	x1, xe := sys.frexp(x);
+	exp := 0;
+	if i := int64(yi); i != 0 {
+		for {
+			if i&1 == 1 {
+				ans *= x1;
+				exp += xe;
+			}
+			i >>= 1;
+			if i == 0 {
+				break;
+			}
+			x1 *= x1;
+			xe <<= 1;
+			if x1 < .5 {
+				x1 += x1;
+				xe--;
 			}
-			return Pow(arg1, temp) * Sqrt(arg1);
 		}
-		return Exp(arg2 * Log(arg1));
 	}
 
-	l := int32(temp);
-	temp = 1;
-	for {
-		if l&1 != 0 {
-			temp = temp*arg1;
-		}
-		l >>= 1;
-		if l == 0 {
-			return temp;
-		}
-		arg1 *= arg1;
+	// ans *= 2^exp
+	// if flip { ans = 1 / ans }
+	// but in the opposite order
+	if flip {
+		ans = 1 / ans;
+		exp = -exp;
 	}
-	panic("unreachable")
+	return sys.ldexp(ans, exp);
 }
+
diff --git a/src/lib/math/sin.go b/src/lib/math/sin.go
index 635e60d21..57de55913 100644
--- a/src/lib/math/sin.go
+++ b/src/lib/math/sin.go
@@ -4,6 +4,9 @@
 
 package math
 
+/*
+	Coefficients are #3370 from Hart & Cheney (18.80D).
+*/
 const
 (
 	p0	=  .1357884097877375669092680e8;
@@ -15,6 +18,7 @@ const
 	q1	=  .4081792252343299749395779e6;
 	q2	=  .9463096101538208180571257e4;
 	q3	=  .1326534908786136358911494e3;
+
         piu2	=  .6366197723675813430755350e0;	// 2/pi
 )