Average Error: 58.0 → 28.3
Time: 34.4s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le 1.568700811656534 \cdot 10^{-11}:\\ \;\;\;\;\left(-2 \cdot im + \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\cos re}{e^{im}} \cdot \left(\frac{\cos re}{e^{im}} \cdot \frac{\cos re}{e^{im}}\right)} - e^{im} \cdot \cos re\right) \cdot 0.5\\ \end{array}\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\begin{array}{l}
\mathbf{if}\;im \le 1.568700811656534 \cdot 10^{-11}:\\
\;\;\;\;\left(-2 \cdot im + \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r6962925 = 0.5;
        double r6962926 = re;
        double r6962927 = cos(r6962926);
        double r6962928 = r6962925 * r6962927;
        double r6962929 = 0.0;
        double r6962930 = im;
        double r6962931 = r6962929 - r6962930;
        double r6962932 = exp(r6962931);
        double r6962933 = exp(r6962930);
        double r6962934 = r6962932 - r6962933;
        double r6962935 = r6962928 * r6962934;
        return r6962935;
}
↓
double f(double re, double im) {
        double r6962936 = im;
        double r6962937 = 1.568700811656534e-11;
        bool r6962938 = r6962936 <= r6962937;
        double r6962939 = -2.0;
        double r6962940 = r6962939 * r6962936;
        double r6962941 = r6962936 * r6962936;
        double r6962942 = r6962936 * r6962941;
        double r6962943 = -0.3333333333333333;
        double r6962944 = r6962942 * r6962943;
        double r6962945 = r6962940 + r6962944;
        double r6962946 = 0.5;
        double r6962947 = r6962945 * r6962946;
        double r6962948 = re;
        double r6962949 = cos(r6962948);
        double r6962950 = exp(r6962936);
        double r6962951 = r6962949 / r6962950;
        double r6962952 = r6962951 * r6962951;
        double r6962953 = r6962951 * r6962952;
        double r6962954 = cbrt(r6962953);
        double r6962955 = r6962950 * r6962949;
        double r6962956 = r6962954 - r6962955;
        double r6962957 = r6962956 * r6962946;
        double r6962958 = r6962938 ? r6962947 : r6962957;
        return r6962958;
}

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
Herbie28.3
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if im < 1.568700811656534e-11

    1. Initial program 59.0

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

    2. Simplified59.1

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

    3. Taylor expanded around 0 31.7

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

    4. Simplified31.7

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

    5. Taylor expanded around 0 28.7

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

    if 1.568700811656534e-11 < im

    1. Initial program 12.3

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

    2. Simplified13.0

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

    4. Applied add-cbrt-cube13.1

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

    5. Applied add-cbrt-cube13.4

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

    6. Applied cbrt-undiv13.0

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

    7. Simplified13.1

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

  4. Final simplification28.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le 1.568700811656534 \cdot 10^{-11}:\\ \;\;\;\;\left(-2 \cdot im + \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\cos re}{e^{im}} \cdot \left(\frac{\cos re}{e^{im}} \cdot \frac{\cos re}{e^{im}}\right)} - e^{im} \cdot \cos re\right) \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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))))