math.sin on complex, imaginary part

Average Error: 57.9 → 58.0

Time: 12.0s

Precision: binary64

Cost: 52800

Math

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

↓

\[\left(0.5 \cdot \left(3 \cdot {\left(\sqrt[3]{\cos re \cdot 0.3333333333333333}\right)}^{3}\right)\right) \cdot \frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (*
  (* 0.5 (* 3.0 (pow (cbrt (* (cos re) 0.3333333333333333)) 3.0)))
  (/
   (- (exp (* im -3.0)) (exp (* 3.0 im)))
   (+ 1.0 (+ (exp (* im -2.0)) (pow (exp 2.0) 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 (0.5 * (3.0 * pow(cbrt((cos(re) * 0.3333333333333333)), 3.0))) * ((exp((im * -3.0)) - exp((3.0 * im))) / (1.0 + (exp((im * -2.0)) + pow(exp(2.0), im))));
}

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 (0.5 * (3.0 * Math.pow(Math.cbrt((Math.cos(re) * 0.3333333333333333)), 3.0))) * ((Math.exp((im * -3.0)) - Math.exp((3.0 * im))) / (1.0 + (Math.exp((im * -2.0)) + Math.pow(Math.exp(2.0), 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(Float64(0.5 * Float64(3.0 * (cbrt(Float64(cos(re) * 0.3333333333333333)) ^ 3.0))) * Float64(Float64(exp(Float64(im * -3.0)) - exp(Float64(3.0 * im))) / Float64(1.0 + Float64(exp(Float64(im * -2.0)) + (exp(2.0) ^ 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[(0.5 * N[(3.0 * N[Power[N[Power[N[(N[Cos[re], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[N[(im * -3.0), $MachinePrecision]], $MachinePrecision] - N[Exp[N[(3.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Exp[N[(im * -2.0), $MachinePrecision]], $MachinePrecision] + N[Power[N[Exp[2.0], $MachinePrecision], im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	57.9
Target	0.2
Herbie	58.0

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

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. Applied egg-rr 58.1

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\frac{{\left(e^{-im}\right)}^{3} - {\left(e^{im}\right)}^{3}}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} + e^{-im} \cdot e^{im}\right)}} \]

4. Simplified 58.0

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)}} \]
   Proof
   (/.f64 (-.f64 (exp.f64 (*.f64 im -3)) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (*.f64 im (Rewrite<= metadata-eval (+.f64 -2 -1)))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (*.f64 im (+.f64 (Rewrite<= metadata-eval (+.f64 -1 -1)) -1))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 im (+.f64 -1 -1)) (*.f64 im -1)))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 im) (*.f64 -1 im))) (*.f64 im -1))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (+.f64 (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 im)) (*.f64 -1 im)) (*.f64 im -1))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (+.f64 (+.f64 (neg.f64 im) (Rewrite<= neg-mul-1_binary64 (neg.f64 im))) (*.f64 im -1))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (+.f64 (+.f64 (neg.f64 im) (neg.f64 im)) (Rewrite<= *-commutative_binary64 (*.f64 -1 im)))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (exp.f64 (+.f64 (+.f64 (neg.f64 im) (neg.f64 im)) (Rewrite<= neg-mul-1_binary64 (neg.f64 im)))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 (+.f64 (neg.f64 im) (neg.f64 im))) (exp.f64 (neg.f64 im)))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 2 points increase in error, 1 points decrease in error
   (/.f64 (-.f64 (*.f64 (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im)))) (exp.f64 (neg.f64 im))) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 2 points increase in error, 3 points decrease in error
   (/.f64 (-.f64 (Rewrite<= unpow3_binary64 (pow.f64 (exp.f64 (neg.f64 im)) 3)) (exp.f64 (*.f64 3 im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 2 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (exp.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) im))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (exp.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 im (*.f64 2 im))))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (exp.f64 (+.f64 im (Rewrite<= count-2_binary64 (+.f64 im im))))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 im) (exp.f64 (+.f64 im im))))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 1 points increase in error, 1 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (*.f64 (exp.f64 im) (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 im) (exp.f64 im))))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 3 points increase in error, 2 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (Rewrite<= cube-mult_binary64 (pow.f64 (exp.f64 im) 3))) (+.f64 1 (+.f64 (exp.f64 (*.f64 im -2)) (pow.f64 (exp.f64 2) im)))): 1 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (exp.f64 (*.f64 im (Rewrite<= metadata-eval (+.f64 -1 -1)))) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (exp.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 im) (*.f64 -1 im)))) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (exp.f64 (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 im)) (*.f64 -1 im))) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (exp.f64 (+.f64 (neg.f64 im) (Rewrite<= neg-mul-1_binary64 (neg.f64 im)))) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im)))) (pow.f64 (exp.f64 2) im)))): 0 points increase in error, 1 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im))) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 2 im)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im))) (exp.f64 (Rewrite<= count-2_binary64 (+.f64 im im)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 1 (+.f64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im))) (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 im) (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im))) (*.f64 (exp.f64 im) (exp.f64 im))) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 (+.f64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im))) (*.f64 (exp.f64 im) (exp.f64 im))) (Rewrite<= lft-mult-inverse_binary64 (*.f64 (/.f64 1 (exp.f64 im)) (exp.f64 im))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (+.f64 (+.f64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im))) (*.f64 (exp.f64 im) (exp.f64 im))) (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 im))) (exp.f64 im)))): 2 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 (exp.f64 (neg.f64 im)) 3) (pow.f64 (exp.f64 im) 3)) (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 (neg.f64 im))) (+.f64 (*.f64 (exp.f64 im) (exp.f64 im)) (*.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))))): 0 points increase in error, 2 points decrease in error

5. Applied egg-rr 58.0

   \[\leadsto \left(0.5 \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\cos re}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos re}}\right)\right)}\right) \cdot \frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \]

6. Simplified 58.0

   \[\leadsto \left(0.5 \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{e^{\cos re}}\right)\right)}\right) \cdot \frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \]
   Proof
   (*.f64 3 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 re))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) (log.f64 (cbrt.f64 (exp.f64 (cos.f64 re))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 re))))) (log.f64 (cbrt.f64 (exp.f64 (cos.f64 re)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= log-pow_binary64 (log.f64 (pow.f64 (cbrt.f64 (exp.f64 (cos.f64 re))) 2))) (log.f64 (cbrt.f64 (exp.f64 (cos.f64 re))))): 36 points increase in error, 25 points decrease in error

7. Applied egg-rr 58.0

   \[\leadsto \left(0.5 \cdot \left(3 \cdot \color{blue}{{\left(\sqrt[3]{\cos re \cdot 0.3333333333333333}\right)}^{3}}\right)\right) \cdot \frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \]

8. Final simplification 58.0

   \[\leadsto \left(0.5 \cdot \left(3 \cdot {\left(\sqrt[3]{\cos re \cdot 0.3333333333333333}\right)}^{3}\right)\right) \cdot \frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \]

Alternatives

Alternative 1
Error	58.0
Cost	52736

\[\frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \cdot \left(0.5 \cdot \log \left(\left(1 + e^{\cos re}\right) + -1\right)\right) \]

Alternative 2
Error	58.0
Cost	40192

\[\frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \cdot \left(0.5 \cdot \left(3 \cdot \left(\left(\cos re \cdot 0.3333333333333333 + 1\right) + -1\right)\right)\right) \]

Alternative 3
Error	58.0
Cost	39936

\[\frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \cdot \left(0.5 \cdot \left(3 \cdot \left(\cos re \cdot 0.3333333333333333\right)\right)\right) \]

Alternative 4
Error	58.0
Cost	39936

\[\frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \cdot \left(0.5 \cdot \left(\left(\cos re + 1\right) + -1\right)\right) \]

Alternative 5
Error	58.0
Cost	39680

\[\frac{e^{im \cdot -3} - e^{3 \cdot im}}{1 + \left(e^{im \cdot -2} + {\left(e^{2}\right)}^{im}\right)} \cdot \left(0.5 \cdot \cos re\right) \]

Alternative 6
Error	58.0
Cost	33408

\[\left(0.5 \cdot \left(-1 + \left(\left(\cos re + 2\right) + -1\right)\right)\right) \cdot \frac{e^{im \cdot -2} - e^{im + im}}{e^{-im} + e^{im}} \]

Alternative 7
Error	58.0
Cost	33024

\[\left(0.5 \cdot \cos re\right) \cdot \frac{e^{im \cdot -2} - e^{im + im}}{e^{-im} + e^{im}} \]

Alternative 8
Error	58.4
Cost	32512

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\right)\right) \]

Alternative 9
Error	58.0
Cost	19968

\[\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \left(-2 + \left(\cos re + 2\right)\right)\right) \]

Alternative 10
Error	57.9
Cost	19712

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

Reproduce

herbie shell --seed 2022334 
(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))))