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math/cmplx: use signed zero to correct branch cuts
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Branch cuts for the elementary complex functions along real or imaginary axes
should be resolved in floating point calculations by one-sided continuity with
signed zero as described in:

"Branch Cuts for Complex Elementary Functions or Much Ado About Nothing's Sign Bit"
W. Kahan

Available at: https://people.freebsd.org/~das/kahan86branch.pdf

And as described in the C99 standard which is claimed as the original cephes source.

Sqrt did not return the correct branch when imag(x) == 0. The branch is now
determined by sign(imag(x)).  This incorrect branch choice was affecting the behavior
of the Trigonometric/Hyperbolic functions that use Sqrt in intermediate calculations.

Asin, Asinh and Atan had spurious domain checks, whereas the functions should be valid
over the whole complex plane with appropriate branch cuts.

Fixes golang#6888

Change-Id: I9b1278af54f54bfb4208276ae345bbd3ddf3ec83
Reviewed-on: https://go-review.googlesource.com/46492
Run-TryBot: Brad Fitzpatrick <[email protected]>
TryBot-Result: Gobot Gobot <[email protected]>
Reviewed-by: Russ Cox <[email protected]>
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@bmkessler @bradfitz
bmkessler authored and bradfitz committed Nov 27, 2017
1 parent 9dbeb92 commit 802a8f8
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Showing 3 changed files with 71 additions and 18 deletions.
19 changes: 4 additions & 15 deletions src/math/cmplx/asin.go
View file
Original file line number Diff line number Diff line change
Expand Up @@ -49,11 +49,8 @@ import "math"

// Asin returns the inverse sine of x.
func Asin(x complex128) complex128 {
if imag(x) == 0 {
if math.Abs(real(x)) > 1 {
return complex(math.Pi/2, 0) // DOMAIN error
}
return complex(math.Asin(real(x)), 0)
if imag(x) == 0 && math.Abs(real(x)) <= 1 {
return complex(math.Asin(real(x)), imag(x))
}
ct := complex(-imag(x), real(x)) // i * x
xx := x * x
Expand All @@ -65,12 +62,8 @@ func Asin(x complex128) complex128 {

// Asinh returns the inverse hyperbolic sine of x.
func Asinh(x complex128) complex128 {
// TODO check range
if imag(x) == 0 {
if math.Abs(real(x)) > 1 {
return complex(math.Pi/2, 0) // DOMAIN error
}
return complex(math.Asinh(real(x)), 0)
if imag(x) == 0 && math.Abs(real(x)) <= 1 {
return complex(math.Asinh(real(x)), imag(x))
}
xx := x * x
x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
Expand Down Expand Up @@ -140,10 +133,6 @@ func Acosh(x complex128) complex128 {

// Atan returns the inverse tangent of x.
func Atan(x complex128) complex128 {
if real(x) == 0 && imag(x) > 1 {
return NaN()
}

x2 := real(x) * real(x)
a := 1 - x2 - imag(x)*imag(x)
if a == 0 {
Expand Down
63 changes: 63 additions & 0 deletions src/math/cmplx/cmath_test.go
View file
Original file line number Diff line number Diff line change
Expand Up @@ -435,6 +435,24 @@ var tanhSC = []complex128{
NaN(),
}

// branch cut continuity checks
// points on each axis at |z| > 1 are checked for one-sided continuity from both the positive and negative side
// all possible branch cuts for the elementary functions are at one of these points

var zero = 0.0
var eps = 1.0 / (1 << 53)

var branchPoints = [][2]complex128{
{complex(2.0, zero), complex(2.0, eps)},
{complex(2.0, -zero), complex(2.0, -eps)},
{complex(-2.0, zero), complex(-2.0, eps)},
{complex(-2.0, -zero), complex(-2.0, -eps)},
{complex(zero, 2.0), complex(eps, 2.0)},
{complex(-zero, 2.0), complex(-eps, 2.0)},
{complex(zero, -2.0), complex(eps, -2.0)},
{complex(-zero, -2.0), complex(-eps, -2.0)},
}

// functions borrowed from pkg/math/all_test.go
func tolerance(a, b, e float64) bool {
d := a - b
Expand Down Expand Up @@ -508,6 +526,11 @@ func TestAcos(t *testing.T) {
t.Errorf("Acos(%g) = %g, want %g", vcAcosSC[i], f, acosSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Acos(pt[0]), Acos(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Acos(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestAcosh(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand All @@ -520,6 +543,11 @@ func TestAcosh(t *testing.T) {
t.Errorf("Acosh(%g) = %g, want %g", vcAcoshSC[i], f, acoshSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Acosh(pt[0]), Acosh(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Acosh(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestAsin(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand All @@ -532,6 +560,11 @@ func TestAsin(t *testing.T) {
t.Errorf("Asin(%g) = %g, want %g", vcAsinSC[i], f, asinSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Asin(pt[0]), Asin(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Asin(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestAsinh(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand All @@ -544,6 +577,11 @@ func TestAsinh(t *testing.T) {
t.Errorf("Asinh(%g) = %g, want %g", vcAsinhSC[i], f, asinhSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Asinh(pt[0]), Asinh(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Asinh(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestAtan(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand All @@ -556,6 +594,11 @@ func TestAtan(t *testing.T) {
t.Errorf("Atan(%g) = %g, want %g", vcAtanSC[i], f, atanSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Atan(pt[0]), Atan(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Atan(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestAtanh(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand All @@ -568,6 +611,11 @@ func TestAtanh(t *testing.T) {
t.Errorf("Atanh(%g) = %g, want %g", vcAtanhSC[i], f, atanhSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Atanh(pt[0]), Atanh(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Atanh(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestConj(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand Down Expand Up @@ -635,6 +683,11 @@ func TestLog(t *testing.T) {
t.Errorf("Log(%g) = %g, want %g", vcLogSC[i], f, logSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Log(pt[0]), Log(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Log(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestLog10(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand Down Expand Up @@ -685,6 +738,11 @@ func TestPow(t *testing.T) {
t.Errorf("Pow(%g, %g) = %g, want %g", vcPowSC[i][0], vcPowSC[i][0], f, powSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Pow(pt[0], 0.1), Pow(pt[1], 0.1); !cVeryclose(f0, f1) {
t.Errorf("Pow(%g, 0.1) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestRect(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand Down Expand Up @@ -733,6 +791,11 @@ func TestSqrt(t *testing.T) {
t.Errorf("Sqrt(%g) = %g, want %g", vcSqrtSC[i], f, sqrtSC[i])
}
}
for _, pt := range branchPoints {
if f0, f1 := Sqrt(pt[0]), Sqrt(pt[1]); !cVeryclose(f0, f1) {
t.Errorf("Sqrt(%g) not continuous, got %g want %g", pt[0], f0, f1)
}
}
}
func TestTan(t *testing.T) {
for i := 0; i < len(vc); i++ {
Expand Down
7 changes: 4 additions & 3 deletions src/math/cmplx/sqrt.go
View file
Original file line number Diff line number Diff line change
Expand Up @@ -57,13 +57,14 @@ import "math"
// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
func Sqrt(x complex128) complex128 {
if imag(x) == 0 {
// Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero.
if real(x) == 0 {
return complex(0, 0)
return complex(0, imag(x))
}
if real(x) < 0 {
return complex(0, math.Sqrt(-real(x)))
return complex(0, math.Copysign(math.Sqrt(-real(x)), imag(x)))
}
return complex(math.Sqrt(real(x)), 0)
return complex(math.Sqrt(real(x)), imag(x))
}
if real(x) == 0 {
if imag(x) < 0 {
Expand Down

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