Average Error: 58.1 → 29.8
Time: 27.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.2955595299771083 \cdot 10^{+92}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;re \le 7.407221374519094 \cdot 10^{+43}:\\ \;\;\;\;\left(\left(\sqrt[3]{\left(\left(re \cdot re\right) \cdot re\right) \cdot \log \left(e^{\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot im\right)}\right)} - \left(im + im\right)\right) - \frac{1}{3} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}} \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}}\right) \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}}\right)\\ \end{array}\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.2955595299771083 \cdot 10^{+92}:\\
\;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\

\mathbf{elif}\;re \le 7.407221374519094 \cdot 10^{+43}:\\
\;\;\;\;\left(\left(\sqrt[3]{\left(\left(re \cdot re\right) \cdot re\right) \cdot \log \left(e^{\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot im\right)}\right)} - \left(im + im\right)\right) - \frac{1}{3} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right) \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}} \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}}\right) \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}}\right)\\

\end{array}
double f(double re, double im) {
        double r3942087 = 0.5;
        double r3942088 = re;
        double r3942089 = cos(r3942088);
        double r3942090 = r3942087 * r3942089;
        double r3942091 = 0.0;
        double r3942092 = im;
        double r3942093 = r3942091 - r3942092;
        double r3942094 = exp(r3942093);
        double r3942095 = exp(r3942092);
        double r3942096 = r3942094 - r3942095;
        double r3942097 = r3942090 * r3942096;
        return r3942097;
}
↓
double f(double re, double im) {
        double r3942098 = re;
        double r3942099 = -4.2955595299771083e+92;
        bool r3942100 = r3942098 <= r3942099;
        double r3942101 = 0.5;
        double r3942102 = cos(r3942098);
        double r3942103 = r3942101 * r3942102;
        double r3942104 = im;
        double r3942105 = -r3942104;
        double r3942106 = exp(r3942105);
        double r3942107 = exp(r3942104);
        double r3942108 = r3942106 - r3942107;
        double r3942109 = r3942103 * r3942108;
        double r3942110 = 7.407221374519094e+43;
        bool r3942111 = r3942098 <= r3942110;
        double r3942112 = r3942098 * r3942098;
        double r3942113 = r3942112 * r3942098;
        double r3942114 = r3942104 * r3942104;
        double r3942115 = r3942114 * r3942104;
        double r3942116 = r3942113 * r3942115;
        double r3942117 = exp(r3942116);
        double r3942118 = log(r3942117);
        double r3942119 = r3942113 * r3942118;
        double r3942120 = cbrt(r3942119);
        double r3942121 = r3942104 + r3942104;
        double r3942122 = r3942120 - r3942121;
        double r3942123 = 0.3333333333333333;
        double r3942124 = r3942123 * r3942115;
        double r3942125 = r3942122 - r3942124;
        double r3942126 = r3942125 * r3942101;
        double r3942127 = r3942102 / r3942107;
        double r3942128 = r3942102 * r3942107;
        double r3942129 = r3942127 - r3942128;
        double r3942130 = cbrt(r3942129);
        double r3942131 = r3942130 * r3942130;
        double r3942132 = r3942131 * r3942130;
        double r3942133 = r3942101 * r3942132;
        double r3942134 = r3942111 ? r3942126 : r3942133;
        double r3942135 = r3942100 ? r3942109 : r3942134;
        return r3942135;
}

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
Herbie29.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -4.2955595299771083e+92

    1. Initial program 58.2

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

    if -4.2955595299771083e+92 < re < 7.407221374519094e+43

    1. Initial program 58.1

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

    2. Simplified58.1

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

    3. Taylor expanded around 0 11.8

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

    4. Simplified11.8

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

    5. Taylor expanded around 0 11.8

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

    6. Simplified11.8

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

    8. Applied add-cbrt-cube11.8

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

    9. Applied add-cbrt-cube11.3

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

    10. Applied cbrt-unprod11.3

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

    11. Applied add-cbrt-cube11.3

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

    12. Applied cbrt-unprod11.3

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

    14. Applied add-log-exp11.2

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

    if 7.407221374519094e+43 < re

    1. Initial program 58.1

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

    2. Simplified58.1

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

    4. Applied add-cube-cbrt58.1

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

  4. Final simplification29.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.2955595299771083 \cdot 10^{+92}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;re \le 7.407221374519094 \cdot 10^{+43}:\\ \;\;\;\;\left(\left(\sqrt[3]{\left(\left(re \cdot re\right) \cdot re\right) \cdot \log \left(e^{\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot im\right)}\right)} - \left(im + im\right)\right) - \frac{1}{3} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}} \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}}\right) \cdot \sqrt[3]{\frac{\cos re}{e^{im}} - \cos re \cdot e^{im}}\right)\\ \end{array}\]

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))))