Average Error: 57.8 → 0.5
Time: 26.8s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le 0.01441673651907451024489770219361162162386:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \end{array}\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\begin{array}{l}
\mathbf{if}\;im \le 0.01441673651907451024489770219361162162386:\\
\;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\

\end{array}
double f(double re, double im) {
        double r117872 = 0.5;
        double r117873 = re;
        double r117874 = cos(r117873);
        double r117875 = r117872 * r117874;
        double r117876 = 0.0;
        double r117877 = im;
        double r117878 = r117876 - r117877;
        double r117879 = exp(r117878);
        double r117880 = exp(r117877);
        double r117881 = r117879 - r117880;
        double r117882 = r117875 * r117881;
        return r117882;
}
↓
double f(double re, double im) {
        double r117883 = im;
        double r117884 = 0.01441673651907451;
        bool r117885 = r117883 <= r117884;
        double r117886 = 0.5;
        double r117887 = re;
        double r117888 = cos(r117887);
        double r117889 = r117886 * r117888;
        double r117890 = 0.3333333333333333;
        double r117891 = 3.0;
        double r117892 = pow(r117883, r117891);
        double r117893 = r117890 * r117892;
        double r117894 = 0.016666666666666666;
        double r117895 = 5.0;
        double r117896 = pow(r117883, r117895);
        double r117897 = r117894 * r117896;
        double r117898 = 2.0;
        double r117899 = r117898 * r117883;
        double r117900 = r117897 + r117899;
        double r117901 = r117893 + r117900;
        double r117902 = -r117901;
        double r117903 = r117889 * r117902;
        double r117904 = -r117883;
        double r117905 = exp(r117904);
        double r117906 = exp(r117883);
        double r117907 = r117905 - r117906;
        double r117908 = r117907 * r117888;
        double r117909 = r117886 * r117908;
        double r117910 = r117885 ? r117903 : r117909;
        return r117910;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.8
Target0.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if im < 0.01441673651907451

    1. Initial program 58.2

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

    2. Taylor expanded around 0 0.5

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

    if 0.01441673651907451 < im

    1. Initial program 0.7

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

    2. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le 0.01441673651907451024489770219361162162386:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 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))))