Average Error: 58.1 → 28.6
Time: 26.8s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos re \le 0.40760495938011426:\\ \;\;\;\;\left(\sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re} \cdot \left(\sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re} \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re}\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\log \left({\left(e^{im}\right)}^{\left(re \cdot re - \frac{1}{3} \cdot \left(im \cdot im\right)\right)}\right) - \left(im + im\right)\right)\\ \end{array}\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\begin{array}{l}
\mathbf{if}\;\cos re \le 0.40760495938011426:\\
\;\;\;\;\left(\sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re} \cdot \left(\sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re} \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re}\right)\right) \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\log \left({\left(e^{im}\right)}^{\left(re \cdot re - \frac{1}{3} \cdot \left(im \cdot im\right)\right)}\right) - \left(im + im\right)\right)\\

\end{array}
double f(double re, double im) {
        double r3678946 = 0.5;
        double r3678947 = re;
        double r3678948 = cos(r3678947);
        double r3678949 = r3678946 * r3678948;
        double r3678950 = 0.0;
        double r3678951 = im;
        double r3678952 = r3678950 - r3678951;
        double r3678953 = exp(r3678952);
        double r3678954 = exp(r3678951);
        double r3678955 = r3678953 - r3678954;
        double r3678956 = r3678949 * r3678955;
        return r3678956;
}
↓
double f(double re, double im) {
        double r3678957 = re;
        double r3678958 = cos(r3678957);
        double r3678959 = 0.40760495938011426;
        bool r3678960 = r3678958 <= r3678959;
        double r3678961 = im;
        double r3678962 = exp(r3678961);
        double r3678963 = r3678958 / r3678962;
        double r3678964 = r3678962 * r3678958;
        double r3678965 = r3678963 - r3678964;
        double r3678966 = cbrt(r3678965);
        double r3678967 = r3678966 * r3678966;
        double r3678968 = r3678966 * r3678967;
        double r3678969 = 0.5;
        double r3678970 = r3678968 * r3678969;
        double r3678971 = r3678957 * r3678957;
        double r3678972 = 0.3333333333333333;
        double r3678973 = r3678961 * r3678961;
        double r3678974 = r3678972 * r3678973;
        double r3678975 = r3678971 - r3678974;
        double r3678976 = pow(r3678962, r3678975);
        double r3678977 = log(r3678976);
        double r3678978 = r3678961 + r3678961;
        double r3678979 = r3678977 - r3678978;
        double r3678980 = r3678969 * r3678979;
        double r3678981 = r3678960 ? r3678970 : r3678980;
        return r3678981;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.2
Herbie28.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos re) < 0.40760495938011426

    1. Initial program 58.3

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

    2. Simplified58.4

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

    4. Applied add-cube-cbrt58.4

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

    if 0.40760495938011426 < (cos re)

    1. Initial program 58.0

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

    2. Simplified58.0

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

    3. Taylor expanded around 0 17.9

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

    4. Simplified17.9

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

    6. Applied add-log-exp17.5

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

    8. Applied add-log-exp16.5

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

    9. Applied exp-to-pow14.7

      \[\leadsto \left(\log \color{blue}{\left({\left(e^{im}\right)}^{\left(re \cdot re - \frac{1}{3} \cdot \left(im \cdot im\right)\right)}\right)} - \left(im + im\right)\right) \cdot 0.5\]
  3. Recombined 2 regimes into one program.

  4. Final simplification28.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \le 0.40760495938011426:\\ \;\;\;\;\left(\sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re} \cdot \left(\sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re} \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re}\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\log \left({\left(e^{im}\right)}^{\left(re \cdot re - \frac{1}{3} \cdot \left(im \cdot im\right)\right)}\right) - \left(im + im\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))))