math.sin on complex, imaginary part

Average Error: 57.8 → 0.8

Time: 9.8s

Precision: binary64

Cost: 20032

Math

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

↓

\[\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (*
  (cos re)
  (+
   (* -0.008333333333333333 (pow im 5.0))
   (- (* -0.16666666666666666 (pow im 3.0)) im))))

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 cos(re) * ((-0.008333333333333333 * pow(im, 5.0)) + ((-0.16666666666666666 * pow(im, 3.0)) - im));
}

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 = cos(re) * (((-0.008333333333333333d0) * (im ** 5.0d0)) + (((-0.16666666666666666d0) * (im ** 3.0d0)) - im))
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.cos(re) * ((-0.008333333333333333 * Math.pow(im, 5.0)) + ((-0.16666666666666666 * Math.pow(im, 3.0)) - im));
}

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.cos(re) * ((-0.008333333333333333 * math.pow(im, 5.0)) + ((-0.16666666666666666 * math.pow(im, 3.0)) - im))

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(cos(re) * Float64(Float64(-0.008333333333333333 * (im ^ 5.0)) + Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im)))
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 = cos(re) * ((-0.008333333333333333 * (im ^ 5.0)) + ((-0.16666666666666666 * (im ^ 3.0)) - im));
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[Cos[re], $MachinePrecision] * N[(N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	57.8
Target	0.3
Herbie	0.8

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

Derivation

1. Initial program 57.8

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

2. Simplified 57.8

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]
   Proof
   (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) -1/2 (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (Rewrite<= metadata-eval (*.f64 1/2 -1)) (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (/.f64 (*.f64 1/2 (Rewrite<= exp-0_binary64 (exp.f64 0))) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (exp.f64 0) (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 (exp.f64 im) (*.f64 1/2 -1) (*.f64 1/2 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 0 im)))))): 1 points increase in error, 5 points decrease in error
   (*.f64 (cos.f64 re) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (exp.f64 im) (*.f64 1/2 -1)) (*.f64 1/2 (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 1/2 -1) (exp.f64 im))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 -1 (exp.f64 im)))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (+.f64 (*.f64 1/2 (Rewrite<= neg-mul-1_binary64 (neg.f64 (exp.f64 im)))) (*.f64 1/2 (exp.f64 (-.f64 0 im))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 (neg.f64 (exp.f64 im)) (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (-.f64 0 im)) (neg.f64 (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= sub-neg_binary64 (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 re) 1/2) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (cos.f64 re))) (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in im around 0 0.8

   \[\leadsto \cos re \cdot \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)\right)} \]

4. Final simplification 0.8

   \[\leadsto \cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) \]

Alternatives

Alternative 1
Error	1.0
Cost	19840

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

Alternative 2
Error	1.0
Cost	13312

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

Alternative 3
Error	1.3
Cost	6656

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

Alternative 4
Error	28.7
Cost	128

\[-im \]

Reproduce

herbie shell --seed 2022328 
(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))))