Average Error: 58.2 → 0.6
Time: 1.3m
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \cos re\right) + \frac{\left(im \cdot 0.5\right) \cdot \left(\cos re \cdot \left(\frac{-1}{27} \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + -8\right)\right)}{\left(\left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot -2 + \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)\right) + 4}\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \cos re\right) + \frac{\left(im \cdot 0.5\right) \cdot \left(\cos re \cdot \left(\frac{-1}{27} \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + -8\right)\right)}{\left(\left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot -2 + \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)\right) + 4}
double f(double re, double im) {
        double r39982906 = 0.5;
        double r39982907 = re;
        double r39982908 = cos(r39982907);
        double r39982909 = r39982906 * r39982908;
        double r39982910 = 0.0;
        double r39982911 = im;
        double r39982912 = r39982910 - r39982911;
        double r39982913 = exp(r39982912);
        double r39982914 = exp(r39982911);
        double r39982915 = r39982913 - r39982914;
        double r39982916 = r39982909 * r39982915;
        return r39982916;
}
↓
double f(double re, double im) {
        double r39982917 = im;
        double r39982918 = 5.0;
        double r39982919 = pow(r39982917, r39982918);
        double r39982920 = -0.016666666666666666;
        double r39982921 = r39982919 * r39982920;
        double r39982922 = 0.5;
        double r39982923 = re;
        double r39982924 = cos(r39982923);
        double r39982925 = r39982922 * r39982924;
        double r39982926 = r39982921 * r39982925;
        double r39982927 = r39982917 * r39982922;
        double r39982928 = -0.037037037037037035;
        double r39982929 = r39982917 * r39982917;
        double r39982930 = r39982917 * r39982929;
        double r39982931 = r39982930 * r39982930;
        double r39982932 = r39982928 * r39982931;
        double r39982933 = -8.0;
        double r39982934 = r39982932 + r39982933;
        double r39982935 = r39982924 * r39982934;
        double r39982936 = r39982927 * r39982935;
        double r39982937 = 0.3333333333333333;
        double r39982938 = r39982929 * r39982937;
        double r39982939 = -2.0;
        double r39982940 = r39982938 * r39982939;
        double r39982941 = r39982938 * r39982938;
        double r39982942 = r39982940 + r39982941;
        double r39982943 = 4.0;
        double r39982944 = r39982942 + r39982943;
        double r39982945 = r39982936 / r39982944;
        double r39982946 = r39982926 + r39982945;
        return r39982946;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  5. Applied add-log-exp0.6

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\frac{-1}{60} \cdot {im}^{5} - \left(\color{blue}{\log \left(e^{\left(\frac{1}{3} \cdot im\right) \cdot im}\right)} + 2\right) \cdot im\right)\]
  6. Using strategy rm

  7. Applied sub-neg0.6

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

  8. Applied distribute-lft-in0.6

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

  9. Simplified0.6

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

  11. Applied flip3--0.6

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

  12. Applied associate-*l/0.6

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

  13. Applied associate-*l/0.6

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

  14. Simplified0.6

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\frac{-1}{60} \cdot {im}^{5}\right) + \frac{\color{blue}{\left(\left(-8 + \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \frac{-1}{27}\right) \cdot \cos re\right) \cdot \left(im \cdot 0.5\right)}}{-2 \cdot -2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) + -2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}\]

  15. Final simplification0.6

    \[\leadsto \left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \cos re\right) + \frac{\left(im \cdot 0.5\right) \cdot \left(\cos re \cdot \left(\frac{-1}{27} \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + -8\right)\right)}{\left(\left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot -2 + \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)\right) + 4}\]

Reproduce

herbie shell --seed 2019128 
(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))))