Average Error: 58.3 → 0.7
Time: 38.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r10720025 = 0.5;
        double r10720026 = re;
        double r10720027 = cos(r10720026);
        double r10720028 = r10720025 * r10720027;
        double r10720029 = 0.0;
        double r10720030 = im;
        double r10720031 = r10720029 - r10720030;
        double r10720032 = exp(r10720031);
        double r10720033 = exp(r10720030);
        double r10720034 = r10720032 - r10720033;
        double r10720035 = r10720028 * r10720034;
        return r10720035;
}
↓
double f(double re, double im) {
        double r10720036 = -0.016666666666666666;
        double r10720037 = im;
        double r10720038 = 5.0;
        double r10720039 = pow(r10720037, r10720038);
        double r10720040 = r10720036 * r10720039;
        double r10720041 = r10720037 + r10720037;
        double r10720042 = r10720040 - r10720041;
        double r10720043 = r10720037 * r10720037;
        double r10720044 = 0.3333333333333333;
        double r10720045 = r10720037 * r10720044;
        double r10720046 = r10720043 * r10720045;
        double r10720047 = r10720042 - r10720046;
        double r10720048 = 0.5;
        double r10720049 = re;
        double r10720050 = cos(r10720049);
        double r10720051 = r10720048 * r10720050;
        double r10720052 = r10720047 * r10720051;
        return r10720052;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.3
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019164 
(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))))