Merge branch 'autopar_rebase2' into autopar_develdevel/autopar_devel

Quickly commit changes in the rebase branch.

1 files changed, 125 insertions, 14 deletions

diff --git a/libgo/go/math/cmplx/tan.go b/libgo/go/math/cmplx/tan.go
index 0243ea0..67a1133 100644
--- a/libgo/go/math/cmplx/tan.go
+++ b/libgo/go/math/cmplx/tan.go
@@ -4,7 +4,10 @@
 
 package cmplx
 
-import "math"
+import (
+	"math"
+	"math/bits"
+)
 
 // The original C code, the long comment, and the constants
 // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
@@ -42,7 +45,7 @@ import "math"
 //            cos 2x  +  cosh 2y
 //
 // On the real axis the denominator is zero at odd multiples
-// of PI/2.  The denominator is evaluated by its Taylor
+// of PI/2. The denominator is evaluated by its Taylor
 // series near these points.
 //
 // ctan(z) = -i ctanh(iz).
@@ -57,6 +60,16 @@ import "math"
 
 // Tan returns the tangent of x.
 func Tan(x complex128) complex128 {
+	switch re, im := real(x), imag(x); {
+	case math.IsInf(im, 0):
+		switch {
+		case math.IsInf(re, 0) || math.IsNaN(re):
+			return complex(math.Copysign(0, re), math.Copysign(1, im))
+		}
+		return complex(math.Copysign(0, math.Sin(2*re)), math.Copysign(1, im))
+	case re == 0 && math.IsNaN(im):
+		return x
+	}
 	d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
 	if math.Abs(d) < 0.25 {
 		d = tanSeries(x)
@@ -81,6 +94,16 @@ func Tan(x complex128) complex128 {
 
 // Tanh returns the hyperbolic tangent of x.
 func Tanh(x complex128) complex128 {
+	switch re, im := real(x), imag(x); {
+	case math.IsInf(re, 0):
+		switch {
+		case math.IsInf(im, 0) || math.IsNaN(im):
+			return complex(math.Copysign(1, re), math.Copysign(0, im))
+		}
+		return complex(math.Copysign(1, re), math.Copysign(0, math.Sin(2*im)))
+	case im == 0 && math.IsNaN(re):
+		return x
+	}
 	d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
 	if d == 0 {
 		return Inf()
@@ -88,22 +111,110 @@ func Tanh(x complex128) complex128 {
 	return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
 }
 
-// Program to subtract nearest integer multiple of PI
+// reducePi reduces the input argument x to the range (-Pi/2, Pi/2].
+// x must be greater than or equal to 0. For small arguments it
+// uses Cody-Waite reduction in 3 float64 parts based on:
+// "Elementary Function Evaluation:  Algorithms and Implementation"
+// Jean-Michel Muller, 1997.
+// For very large arguments it uses Payne-Hanek range reduction based on:
+// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
+// K. C. Ng et al, March 24, 1992.
 func reducePi(x float64) float64 {
+	// reduceThreshold is the maximum value of x where the reduction using
+	// Cody-Waite reduction still gives accurate results. This threshold
+	// is set by t*PIn being representable as a float64 without error
+	// where t is given by t = floor(x * (1 / Pi)) and PIn are the leading partial
+	// terms of Pi. Since the leading terms, PI1 and PI2 below, have 30 and 32
+	// trailing zero bits respectively, t should have less than 30 significant bits.
+	//	t < 1<<30  -> floor(x*(1/Pi)+0.5) < 1<<30 -> x < (1<<30-1) * Pi - 0.5
+	// So, conservatively we can take x < 1<<30.
+	const reduceThreshold float64 = 1 << 30
+	if math.Abs(x) < reduceThreshold {
+		// Use Cody-Waite reduction in three parts.
+		const (
+			// PI1, PI2 and PI3 comprise an extended precision value of PI
+			// such that PI ~= PI1 + PI2 + PI3. The parts are chosen so
+			// that PI1 and PI2 have an approximately equal number of trailing
+			// zero bits. This ensures that t*PI1 and t*PI2 are exact for
+			// large integer values of t. The full precision PI3 ensures the
+			// approximation of PI is accurate to 102 bits to handle cancellation
+			// during subtraction.
+			PI1 = 3.141592502593994      // 0x400921fb40000000
+			PI2 = 1.5099578831723193e-07 // 0x3e84442d00000000
+			PI3 = 1.0780605716316238e-14 // 0x3d08469898cc5170
+		)
+		t := x / math.Pi
+		t += 0.5
+		t = float64(int64(t)) // int64(t) = the multiple
+		return ((x - t*PI1) - t*PI2) - t*PI3
+	}
+	// Must apply Payne-Hanek range reduction
 	const (
-		// extended precision value of PI:
-		DP1 = 3.14159265160560607910e0   // ?? 0x400921fb54000000
-		DP2 = 1.98418714791870343106e-9  // ?? 0x3e210b4610000000
-		DP3 = 1.14423774522196636802e-17 // ?? 0x3c6a62633145c06e
+		mask     = 0x7FF
+		shift    = 64 - 11 - 1
+		bias     = 1023
+		fracMask = 1<<shift - 1
 	)
-	t := x / math.Pi
-	if t >= 0 {
-		t += 0.5
-	} else {
-		t -= 0.5
+	// Extract out the integer and exponent such that,
+	// x = ix * 2 ** exp.
+	ix := math.Float64bits(x)
+	exp := int(ix>>shift&mask) - bias - shift
+	ix &= fracMask
+	ix |= 1 << shift
+
+	// mPi is the binary digits of 1/Pi as a uint64 array,
+	// that is, 1/Pi = Sum mPi[i]*2^(-64*i).
+	// 19 64-bit digits give 1216 bits of precision
+	// to handle the largest possible float64 exponent.
+	var mPi = [...]uint64{
+		0x0000000000000000,
+		0x517cc1b727220a94,
+		0xfe13abe8fa9a6ee0,
+		0x6db14acc9e21c820,
+		0xff28b1d5ef5de2b0,
+		0xdb92371d2126e970,
+		0x0324977504e8c90e,
+		0x7f0ef58e5894d39f,
+		0x74411afa975da242,
+		0x74ce38135a2fbf20,
+		0x9cc8eb1cc1a99cfa,
+		0x4e422fc5defc941d,
+		0x8ffc4bffef02cc07,
+		0xf79788c5ad05368f,
+		0xb69b3f6793e584db,
+		0xa7a31fb34f2ff516,
+		0xba93dd63f5f2f8bd,
+		0x9e839cfbc5294975,
+		0x35fdafd88fc6ae84,
+		0x2b0198237e3db5d5,
+	}
+	// Use the exponent to extract the 3 appropriate uint64 digits from mPi,
+	// B ~ (z0, z1, z2), such that the product leading digit has the exponent -64.
+	// Note, exp >= 50 since x >= reduceThreshold and exp < 971 for maximum float64.
+	digit, bitshift := uint(exp+64)/64, uint(exp+64)%64
+	z0 := (mPi[digit] << bitshift) | (mPi[digit+1] >> (64 - bitshift))
+	z1 := (mPi[digit+1] << bitshift) | (mPi[digit+2] >> (64 - bitshift))
+	z2 := (mPi[digit+2] << bitshift) | (mPi[digit+3] >> (64 - bitshift))
+	// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
+	z2hi, _ := bits.Mul64(z2, ix)
+	z1hi, z1lo := bits.Mul64(z1, ix)
+	z0lo := z0 * ix
+	lo, c := bits.Add64(z1lo, z2hi, 0)
+	hi, _ := bits.Add64(z0lo, z1hi, c)
+	// Find the magnitude of the fraction.
+	lz := uint(bits.LeadingZeros64(hi))
+	e := uint64(bias - (lz + 1))
+	// Clear implicit mantissa bit and shift into place.
+	hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
+	hi >>= 64 - shift
+	// Include the exponent and convert to a float.
+	hi |= e << shift
+	x = math.Float64frombits(hi)
+	// map to (-Pi/2, Pi/2]
+	if x > 0.5 {
+		x--
 	}
-	t = float64(int64(t)) // int64(t) = the multiple
-	return ((x - t*DP1) - t*DP2) - t*DP3
+	return math.Pi * x
 }
 
 // Taylor series expansion for cosh(2y) - cos(2x)