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// Copyright ©2018 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package dual
import (
"fmt"
"math"
"testing"
"gonum.org/v1/gonum/floats/scalar"
)
var formatTests = []struct {
d Number
format string
want string
}{
{d: Number{1.1, 2.1}, format: "%#v", want: "dual.Number{Real:1.1, Emag:2.1}"}, // Bootstrap test.
{d: Number{-1.1, -2.1}, format: "%#v", want: "dual.Number{Real:-1.1, Emag:-2.1}"}, // Bootstrap test.
{d: Number{1.1, 2.1}, format: "%+v", want: "{Real:1.1, Emag:2.1}"},
{d: Number{-1.1, -2.1}, format: "%+v", want: "{Real:-1.1, Emag:-2.1}"},
{d: Number{1, 2}, format: "%v", want: "(1+2ϵ)"},
{d: Number{-1, -2}, format: "%v", want: "(-1-2ϵ)"},
{d: Number{1, 2}, format: "%g", want: "(1+2ϵ)"},
{d: Number{-1, -2}, format: "%g", want: "(-1-2ϵ)"},
{d: Number{1, 2}, format: "%e", want: "(1.000000e+00+2.000000e+00ϵ)"},
{d: Number{-1, -2}, format: "%e", want: "(-1.000000e+00-2.000000e+00ϵ)"},
{d: Number{1, 2}, format: "%E", want: "(1.000000E+00+2.000000E+00ϵ)"},
{d: Number{-1, -2}, format: "%E", want: "(-1.000000E+00-2.000000E+00ϵ)"},
{d: Number{1, 2}, format: "%f", want: "(1.000000+2.000000ϵ)"},
{d: Number{-1, -2}, format: "%f", want: "(-1.000000-2.000000ϵ)"},
}
func TestFormat(t *testing.T) {
t.Parallel()
for _, test := range formatTests {
got := fmt.Sprintf(test.format, test.d)
if got != test.want {
t.Errorf("unexpected result for fmt.Sprintf(%q, %#v): got:%q, want:%q", test.format, test.d, got, test.want)
}
}
}
// First derivatives:
func dSin(x float64) float64 { return math.Cos(x) }
func dCos(x float64) float64 { return -math.Sin(x) }
func dTan(x float64) float64 { return sec(x) * sec(x) }
func dAsin(x float64) float64 { return 1 / math.Sqrt(1-x*x) }
func dAcos(x float64) float64 { return -1 / math.Sqrt(1-x*x) }
func dAtan(x float64) float64 { return 1 / (1 + x*x) }
func dSinh(x float64) float64 { return math.Cosh(x) }
func dCosh(x float64) float64 { return math.Sinh(x) }
func dTanh(x float64) float64 { return sech(x) * sech(x) }
func dAsinh(x float64) float64 { return 1 / math.Sqrt(x*x+1) }
func dAcosh(x float64) float64 { return 1 / (math.Sqrt(x-1) * math.Sqrt(x+1)) }
func dAtanh(x float64) float64 {
switch {
case math.Abs(x) == 1:
return math.NaN()
case math.IsInf(x, 0):
return negZero
}
return 1 / (1 - x*x)
}
func dExp(x float64) float64 { return math.Exp(x) }
func dLog(x float64) float64 {
if x < 0 {
return math.NaN()
}
return 1 / x
}
func dSqrt(x float64) float64 {
// For whatever reason, math.Sqrt(-0) returns -0.
// In this case, that is clearly a wrong approach.
if x == 0 {
return math.Inf(1)
}
return 0.5 / math.Sqrt(x)
}
func dInv(x float64) float64 { return -1 / (x * x) }
// Helpers:
func sec(x float64) float64 { return 1 / math.Cos(x) }
func sech(x float64) float64 { return 1 / math.Cosh(x) }
var negZero = math.Float64frombits(1 << 63)
var dualTests = []struct {
name string
x []float64
fnDual func(x Number) Number
fn func(x float64) float64
dFn func(x float64) float64
}{
{
name: "sin",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Sin,
fn: math.Sin,
dFn: dSin,
},
{
name: "cos",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Cos,
fn: math.Cos,
dFn: dCos,
},
{
name: "tan",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Tan,
fn: math.Tan,
dFn: dTan,
},
{
name: "sinh",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Sinh,
fn: math.Sinh,
dFn: dSinh,
},
{
name: "cosh",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Cosh,
fn: math.Cosh,
dFn: dCosh,
},
{
name: "tanh",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Tanh,
fn: math.Tanh,
dFn: dTanh,
},
{
name: "asin",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Asin,
fn: math.Asin,
dFn: dAsin,
},
{
name: "acos",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Acos,
fn: math.Acos,
dFn: dAcos,
},
{
name: "atan",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Atan,
fn: math.Atan,
dFn: dAtan,
},
{
name: "asinh",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Asinh,
fn: math.Asinh,
dFn: dAsinh,
},
{
name: "acosh",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Acosh,
fn: math.Acosh,
dFn: dAcosh,
},
{
name: "atanh",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Atanh,
fn: math.Atanh,
dFn: dAtanh,
},
{
name: "exp",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Exp,
fn: math.Exp,
dFn: dExp,
},
{
name: "log",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Log,
fn: math.Log,
dFn: dLog,
},
{
name: "inv",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Inv,
fn: func(x float64) float64 { return 1 / x },
dFn: dInv,
},
{
name: "sqrt",
x: []float64{math.NaN(), math.Inf(-1), -3, -2, -1, -0.5, negZero, 0, 0.5, 1, 2, 3, math.Inf(1)},
fnDual: Sqrt,
fn: math.Sqrt,
dFn: dSqrt,
},
{
name: "Fike example fn",
x: []float64{1, 2, 3, 4, 5},
fnDual: func(x Number) Number {
return Mul(
Exp(x),
Inv(Sqrt(
Add(
PowReal(Sin(x), 3),
PowReal(Cos(x), 3)))))
},
fn: func(x float64) float64 {
return math.Exp(x) / math.Sqrt(math.Pow(math.Sin(x), 3)+math.Pow(math.Cos(x), 3))
},
dFn: func(x float64) float64 {
return math.Exp(x) * (3*math.Cos(x) + 5*math.Cos(3*x) + 9*math.Sin(x) + math.Sin(3*x)) /
(8 * math.Pow(math.Pow(math.Sin(x), 3)+math.Pow(math.Cos(x), 3), 1.5))
},
},
}
func TestDual(t *testing.T) {
t.Parallel()
const tol = 1e-15
for _, test := range dualTests {
for _, x := range test.x {
fxDual := test.fnDual(Number{Real: x, Emag: 1})
fx := test.fn(x)
dFx := test.dFn(x)
if !same(fxDual.Real, fx, tol) {
t.Errorf("unexpected %s(%v): got:%v want:%v", test.name, x, fxDual.Real, fx)
}
if !same(fxDual.Emag, dFx, tol) {
t.Errorf("unexpected %s'(%v): got:%v want:%v", test.name, x, fxDual.Emag, dFx)
}
}
}
}
var powRealTests = []struct {
d Number
p float64
want Number
}{
// PowReal(NaN+xϵ, ±0) = 1+NaNϵ for any x
{d: Number{Real: math.NaN(), Emag: 0}, p: 0, want: Number{Real: 1, Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 0}, p: negZero, want: Number{Real: 1, Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 1}, p: 0, want: Number{Real: 1, Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 2}, p: negZero, want: Number{Real: 1, Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 3}, p: 0, want: Number{Real: 1, Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 1}, p: negZero, want: Number{Real: 1, Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 2}, p: 0, want: Number{Real: 1, Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 3}, p: negZero, want: Number{Real: 1, Emag: math.NaN()}},
// PowReal(x, ±0) = 1 for any x
{d: Number{Real: 0, Emag: 0}, p: 0, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: negZero, Emag: 0}, p: negZero, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: math.Inf(1), Emag: 0}, p: 0, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: math.Inf(-1), Emag: 0}, p: negZero, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: 0, Emag: 1}, p: 0, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: negZero, Emag: 1}, p: negZero, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: math.Inf(1), Emag: 1}, p: 0, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: math.Inf(-1), Emag: 1}, p: negZero, want: Number{Real: 1, Emag: 0}},
// PowReal(1+xϵ, y) = (1+xyϵ) for any y
{d: Number{Real: 1, Emag: 0}, p: 0, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: 1, Emag: 0}, p: 1, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: 1, Emag: 0}, p: 2, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: 1, Emag: 0}, p: 3, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: 1, Emag: 1}, p: 0, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: 1, Emag: 1}, p: 1, want: Number{Real: 1, Emag: 1}},
{d: Number{Real: 1, Emag: 1}, p: 2, want: Number{Real: 1, Emag: 2}},
{d: Number{Real: 1, Emag: 1}, p: 3, want: Number{Real: 1, Emag: 3}},
{d: Number{Real: 1, Emag: 2}, p: 0, want: Number{Real: 1, Emag: 0}},
{d: Number{Real: 1, Emag: 2}, p: 1, want: Number{Real: 1, Emag: 2}},
{d: Number{Real: 1, Emag: 2}, p: 2, want: Number{Real: 1, Emag: 4}},
{d: Number{Real: 1, Emag: 2}, p: 3, want: Number{Real: 1, Emag: 6}},
// PowReal(x, 1) = x for any x
{d: Number{Real: 0, Emag: 0}, p: 1, want: Number{Real: 0, Emag: 0}},
{d: Number{Real: negZero, Emag: 0}, p: 1, want: Number{Real: negZero, Emag: 0}},
{d: Number{Real: 0, Emag: 1}, p: 1, want: Number{Real: 0, Emag: 1}},
{d: Number{Real: negZero, Emag: 1}, p: 1, want: Number{Real: negZero, Emag: 1}},
{d: Number{Real: math.NaN(), Emag: 0}, p: 1, want: Number{Real: math.NaN(), Emag: 0}},
{d: Number{Real: math.NaN(), Emag: 1}, p: 1, want: Number{Real: math.NaN(), Emag: 1}},
{d: Number{Real: math.NaN(), Emag: 2}, p: 1, want: Number{Real: math.NaN(), Emag: 2}},
// PowReal(NaN+xϵ, y) = NaN+NaNϵ
{d: Number{Real: math.NaN(), Emag: 0}, p: 2, want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 0}, p: 3, want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 1}, p: 2, want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 1}, p: 3, want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 2}, p: 2, want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: math.NaN(), Emag: 2}, p: 3, want: Number{Real: math.NaN(), Emag: math.NaN()}},
// PowReal(x, NaN) = NaN+NaNϵ
{d: Number{Real: 0, Emag: 0}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 2, Emag: 0}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 3, Emag: 0}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 0, Emag: 1}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 2, Emag: 1}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 3, Emag: 1}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 0, Emag: 2}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 2, Emag: 2}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: 3, Emag: 2}, p: math.NaN(), want: Number{Real: math.NaN(), Emag: math.NaN()}},
// Handled by math.Pow tests:
//
// Pow(±0, y) = ±Inf for y an odd integer < 0
// Pow(±0, -Inf) = +Inf
// Pow(±0, +Inf) = +0
// Pow(±0, y) = +Inf for finite y < 0 and not an odd integer
// Pow(±0, y) = ±0 for y an odd integer > 0
// Pow(±0, y) = +0 for finite y > 0 and not an odd integer
// Pow(-1, ±Inf) = 1
// PowReal(x+0ϵ, +Inf) = +Inf+NaNϵ for |x| > 1
{d: Number{Real: 2, Emag: 0}, p: math.Inf(1), want: Number{Real: math.Inf(1), Emag: math.NaN()}},
{d: Number{Real: 3, Emag: 0}, p: math.Inf(1), want: Number{Real: math.Inf(1), Emag: math.NaN()}},
// PowReal(x+yϵ, +Inf) = +Inf for |x| > 1
{d: Number{Real: 2, Emag: 1}, p: math.Inf(1), want: Number{Real: math.Inf(1), Emag: math.Inf(1)}},
{d: Number{Real: 3, Emag: 1}, p: math.Inf(1), want: Number{Real: math.Inf(1), Emag: math.Inf(1)}},
{d: Number{Real: 2, Emag: 2}, p: math.Inf(1), want: Number{Real: math.Inf(1), Emag: math.Inf(1)}},
{d: Number{Real: 3, Emag: 2}, p: math.Inf(1), want: Number{Real: math.Inf(1), Emag: math.Inf(1)}},
// PowReal(x, -Inf) = +0+NaNϵ for |x| > 1
{d: Number{Real: 2, Emag: 0}, p: math.Inf(-1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 3, Emag: 0}, p: math.Inf(-1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 2, Emag: 1}, p: math.Inf(-1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 3, Emag: 1}, p: math.Inf(-1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 2, Emag: 2}, p: math.Inf(-1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 3, Emag: 2}, p: math.Inf(-1), want: Number{Real: 0, Emag: math.NaN()}},
// PowReal(x+yϵ, +Inf) = +0+NaNϵ for |x| < 1
{d: Number{Real: 0.1, Emag: 0}, p: math.Inf(1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 0.1, Emag: 0.1}, p: math.Inf(1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 0.2, Emag: 0.2}, p: math.Inf(1), want: Number{Real: 0, Emag: math.NaN()}},
{d: Number{Real: 0.5, Emag: 0.5}, p: math.Inf(1), want: Number{Real: 0, Emag: math.NaN()}},
// PowReal(x+0ϵ, -Inf) = +Inf+NaNϵ for |x| < 1
{d: Number{Real: 0.1, Emag: 0}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.NaN()}},
{d: Number{Real: 0.2, Emag: 0}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.NaN()}},
// PowReal(x, -Inf) = +Inf-Infϵ for |x| < 1
{d: Number{Real: 0.1, Emag: 0.1}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
{d: Number{Real: 0.2, Emag: 0.1}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
{d: Number{Real: 0.1, Emag: 0.2}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
{d: Number{Real: 0.2, Emag: 0.2}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
{d: Number{Real: 0.1, Emag: 1}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
{d: Number{Real: 0.2, Emag: 1}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
{d: Number{Real: 0.1, Emag: 2}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
{d: Number{Real: 0.2, Emag: 2}, p: math.Inf(-1), want: Number{Real: math.Inf(1), Emag: math.Inf(-1)}},
// Handled by math.Pow tests:
//
// Pow(+Inf, y) = +Inf for y > 0
// Pow(+Inf, y) = +0 for y < 0
// Pow(-Inf, y) = Pow(-0, -y)
// PowReal(x, y) = NaN+NaNϵ for finite x < 0 and finite non-integer y
{d: Number{Real: -1, Emag: -1}, p: 0.5, want: Number{Real: math.NaN(), Emag: math.NaN()}},
{d: Number{Real: -1, Emag: 2}, p: 0.5, want: Number{Real: math.NaN(), Emag: math.NaN()}},
}
func TestPowReal(t *testing.T) {
t.Parallel()
const tol = 1e-15
for _, test := range powRealTests {
got := PowReal(test.d, test.p)
if !sameDual(got, test.want, tol) {
t.Errorf("unexpected PowReal(%v, %v): got:%v want:%v", test.d, test.p, got, test.want)
}
}
}
func sameDual(a, b Number, tol float64) bool {
return same(a.Real, b.Real, tol) && same(a.Emag, b.Emag, tol)
}
func same(a, b, tol float64) bool {
return (math.IsNaN(a) && math.IsNaN(b)) || scalar.EqualWithinAbsOrRel(a, b, tol, tol)
}
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