math.sin on complex, imaginary part

Average Error: 57.9 → 0.9

Time: 10.3s

Precision: binary64

Cost: 19840

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

↓

\[{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

↓

(FPCore (re im)
 :precision binary64
 (- (* (pow im 3.0) (* (cos re) -0.16666666666666666)) (* im (cos re))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

↓

double code(double re, double im) {
	return (pow(im, 3.0) * (cos(re) * -0.16666666666666666)) - (im * cos(re));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((im ** 3.0d0) * (cos(re) * (-0.16666666666666666d0))) - (im * cos(re))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

↓

public static double code(double re, double im) {
	return (Math.pow(im, 3.0) * (Math.cos(re) * -0.16666666666666666)) - (im * Math.cos(re));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

↓

def code(re, im):
	return (math.pow(im, 3.0) * (math.cos(re) * -0.16666666666666666)) - (im * math.cos(re))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

↓

function code(re, im)
	return Float64(Float64((im ^ 3.0) * Float64(cos(re) * -0.16666666666666666)) - Float64(im * cos(re)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

↓

function tmp = code(re, im)
	tmp = ((im ^ 3.0) * (cos(re) * -0.16666666666666666)) - (im * cos(re));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	57.9
Target	0.2
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \]

Derivation

1. Initial program 57.9

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

2. Simplified 57.9

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   Proof
   (*.f64 (*.f64 1/2 (cos.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 1/2 (cos.f64 re)) (-.f64 (exp.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 im))) (exp.f64 im))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in im around 0 0.9

   \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]

4. Simplified 0.9

   \[\leadsto \color{blue}{{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re} \]
   Proof
   (-.f64 (*.f64 (pow.f64 im 3) (*.f64 (cos.f64 re) -1/6)) (*.f64 im (cos.f64 re))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (pow.f64 im 3) (*.f64 (cos.f64 re) -1/6)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 re) im))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 im 3) (cos.f64 re)) -1/6)) (*.f64 (cos.f64 re) im)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/6 (*.f64 (pow.f64 im 3) (cos.f64 re)))) (*.f64 (cos.f64 re) im)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1/6 (*.f64 (pow.f64 im 3) (cos.f64 re))) (neg.f64 (*.f64 (cos.f64 re) im)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 -1/6 (*.f64 (pow.f64 im 3) (cos.f64 re))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (cos.f64 re) im)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 (cos.f64 re) im)) (*.f64 -1/6 (*.f64 (pow.f64 im 3) (cos.f64 re))))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.9

   \[\leadsto {im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re \]

Alternatives

Alternative 1
Error	0.9
Cost	13312

\[\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

Alternative 2
Error	1.3
Cost	6656

\[\cos re \cdot \left(-im\right) \]

Alternative 3
Error	28.1
Cost	576

\[im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right) \]

Alternative 4
Error	28.3
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2022335 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))