Average Error: 58.0 → 0.7
Time: 33.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\left(im + im\right) + \frac{1}{60} \cdot {im}^{5}\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\left(im + im\right) + \frac{1}{60} \cdot {im}^{5}\right)\right)
double f(double re, double im) {
        double r11737773 = 0.5;
        double r11737774 = re;
        double r11737775 = cos(r11737774);
        double r11737776 = r11737773 * r11737775;
        double r11737777 = 0.0;
        double r11737778 = im;
        double r11737779 = r11737777 - r11737778;
        double r11737780 = exp(r11737779);
        double r11737781 = exp(r11737778);
        double r11737782 = r11737780 - r11737781;
        double r11737783 = r11737776 * r11737782;
        return r11737783;
}
↓
double f(double re, double im) {
        double r11737784 = 0.5;
        double r11737785 = re;
        double r11737786 = cos(r11737785);
        double r11737787 = r11737784 * r11737786;
        double r11737788 = im;
        double r11737789 = r11737788 * r11737788;
        double r11737790 = r11737788 * r11737789;
        double r11737791 = -0.3333333333333333;
        double r11737792 = r11737790 * r11737791;
        double r11737793 = r11737788 + r11737788;
        double r11737794 = 0.016666666666666666;
        double r11737795 = 5.0;
        double r11737796 = pow(r11737788, r11737795);
        double r11737797 = r11737794 * r11737796;
        double r11737798 = r11737793 + r11737797;
        double r11737799 = r11737792 - r11737798;
        double r11737800 = r11737787 * r11737799;
        return r11737800;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 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))))