Average Error: 58.0 → 0.7
Time: 35.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} + \frac{-1}{60} \cdot {im}^{5}\right) - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} + \frac{-1}{60} \cdot {im}^{5}\right) - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r8919908 = 0.5;
        double r8919909 = re;
        double r8919910 = cos(r8919909);
        double r8919911 = r8919908 * r8919910;
        double r8919912 = 0.0;
        double r8919913 = im;
        double r8919914 = r8919912 - r8919913;
        double r8919915 = exp(r8919914);
        double r8919916 = exp(r8919913);
        double r8919917 = r8919915 - r8919916;
        double r8919918 = r8919911 * r8919917;
        return r8919918;
}
↓
double f(double re, double im) {
        double r8919919 = im;
        double r8919920 = r8919919 * r8919919;
        double r8919921 = r8919919 * r8919920;
        double r8919922 = -0.3333333333333333;
        double r8919923 = r8919921 * r8919922;
        double r8919924 = -0.016666666666666666;
        double r8919925 = 5.0;
        double r8919926 = pow(r8919919, r8919925);
        double r8919927 = r8919924 * r8919926;
        double r8919928 = r8919923 + r8919927;
        double r8919929 = r8919919 + r8919919;
        double r8919930 = r8919928 - r8919929;
        double r8919931 = 0.5;
        double r8919932 = re;
        double r8919933 = cos(r8919932);
        double r8919934 = r8919931 * r8919933;
        double r8919935 = r8919930 * r8919934;
        return r8919935;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.0
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.7

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  3. Simplified0.7

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left({im}^{5} \cdot \frac{-1}{60} + \left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) - \left(im + im\right)\right)}\]
  4. Using strategy rm

  5. Applied *-commutative0.7

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left({im}^{5} \cdot \frac{-1}{60} + \color{blue}{\frac{-1}{3} \cdot \left(\left(im \cdot im\right) \cdot im\right)}\right) - \left(im + im\right)\right)\]

  6. Final simplification0.7

    \[\leadsto \left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} + \frac{-1}{60} \cdot {im}^{5}\right) - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))