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power.go
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power.go
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// Copyright 2014 The Go 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 ivyshims
import (
"math/big"
)
func power(c Context, u, v Value) Value {
// Because of the promotions done in binary.go, if one
// argument is complex, they both are.
if _, ok := u.(Complex); ok {
return complexPower(c, u.(Complex), v.(Complex)).shrink()
}
z := floatPower(c, floatSelf(c, u), floatSelf(c, v))
return BigFloat{z}.shrink()
}
func exp(c Context, v Value) Value {
if u, ok := v.(Complex); ok {
if !isZero(u.imag) {
return expComplex(c, u)
}
v = u.real
}
z := exponential(c.Config(), floatSelf(c, v).Float)
return BigFloat{z}.shrink()
}
// expComplex returns e**v where v is Complex.
func expComplex(c Context, v Complex) Value {
// Use the Euler formula: e**ix == cos x + i sin x.
// Thus e**(x+iy) == e**x * (cos y + i sin y).
// First turn v into (a + bi) where a and b are big.Floats.
x := floatSelf(c, v.real).Float
y := floatSelf(c, v.imag).Float
eToX := exponential(c.Config(), x)
cosY := floatCos(c, y)
sinY := floatSin(c, y)
return newComplex(BigFloat{cosY.Mul(cosY, eToX)}, BigFloat{sinY.Mul(sinY, eToX)})
}
// floatPower computes bx to the power of bexp.
func floatPower(c Context, bx, bexp BigFloat) *big.Float {
x := bx.Float
fexp := newFloat(c).Set(bexp.Float)
positive := true
conf := c.Config()
switch fexp.Sign() {
case 0:
return newFloat(c).SetInt64(1)
case -1:
if x.Sign() == 0 {
Errorf("negative exponent of zero")
}
positive = false
fexp = c.EvalUnary("-", bexp).toType("**", conf, bigFloatType).(BigFloat).Float
}
// Easy cases.
switch {
case x.Cmp(floatOne) == 0, x.Sign() == 0:
return x
case fexp.Cmp(floatHalf) == 0:
z := floatSqrt(c, x)
if !positive {
z = z.Quo(floatOne, z)
}
return z
}
isInt := true
exp, acc := fexp.Int64() // No point in doing *big.Ints now. TODO?
if acc != big.Exact {
isInt = false
}
// Integer part.
z := integerPower(c, x, exp)
// Fractional part.
if !isInt {
frac := fexp.Sub(fexp, newFloat(c).SetInt64(exp))
// x**frac is e**(frac*log x)
logx := floatLog(c, x)
frac.Mul(frac, logx)
z.Mul(z, exponential(c.Config(), frac))
}
if !positive {
z.Quo(floatOne, z)
}
return z
}
// exponential computes exp(x) using the Taylor series. It converges quickly
// since we call it with only small values of x.
func exponential(conf *Config, x *big.Float) *big.Float {
// The Taylor series for e**x, exp(x), is 1 + x + x²/2! + x³/3! ...
xN := newF(conf).Set(x)
term := newF(conf)
n := newF(conf)
nFactorial := newF(conf).SetUint64(1)
z := newF(conf).SetInt64(1)
for loop := newLoop(conf, "exponential", x, 10); ; { // Big exponentials converge slowly.
term.Set(xN)
term.Quo(term, nFactorial)
z.Add(z, term)
if loop.done(z) {
break
}
// Advance x**index (multiply by x).
xN.Mul(xN, x)
// Advance n, n!.
nFactorial.Mul(nFactorial, n.SetUint64(loop.i+1))
}
return z
}
// integerPower returns x**exp where exp is an int64 of size <= intBits.
func integerPower(c Context, x *big.Float, exp int64) *big.Float {
z := newFloat(c).SetInt64(1)
y := newFloat(c).Set(x)
// For each loop, we compute xⁿ where n is a power of two.
for exp > 0 {
if exp&1 == 1 {
// This bit contributes. Multiply it into the result.
z.Mul(z, y)
}
y.Mul(y, y)
exp >>= 1
}
return z
}
// complexIntegerPower returns x**exp where exp is an int64 of size <= intBits.
func complexIntegerPower(c Context, v Complex, exp int64) Complex {
z := newComplex(Int(1), Int(0))
y := newComplex(v.real, v.imag)
// For each loop, we compute xⁿ where n is a power of two.
for exp > 0 {
if exp&1 == 1 {
// This bit contributes. Multiply it into the result.
z = z.mul(c, y)
}
y = y.mul(c, y)
exp >>= 1
}
return z
}
// complexPower computes v to the power of exp.
func complexPower(c Context, v, exp Complex) Value {
if isZero(exp.imag) {
if i, ok := exp.real.(Int); ok {
switch {
case i == 0:
return Int(1)
case i == 1:
return v
case i < 0:
return complexIntegerPower(c, v, -int64(i)).recip(c)
default:
return complexIntegerPower(c, v, int64(i))
}
}
}
return expComplex(c, complexLog(c, v).mul(c, exp))
}