Average Error: 58.0 → 0.7
Time: 33.3s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) + \left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\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(-\left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) + \left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r9927853 = 0.5;
        double r9927854 = re;
        double r9927855 = cos(r9927854);
        double r9927856 = r9927853 * r9927855;
        double r9927857 = 0.0;
        double r9927858 = im;
        double r9927859 = r9927857 - r9927858;
        double r9927860 = exp(r9927859);
        double r9927861 = exp(r9927858);
        double r9927862 = r9927860 - r9927861;
        double r9927863 = r9927856 * r9927862;
        return r9927863;
}
↓
double f(double re, double im) {
        double r9927864 = 0.5;
        double r9927865 = re;
        double r9927866 = cos(r9927865);
        double r9927867 = r9927864 * r9927866;
        double r9927868 = im;
        double r9927869 = r9927868 + r9927868;
        double r9927870 = -r9927869;
        double r9927871 = 0.016666666666666666;
        double r9927872 = 5.0;
        double r9927873 = pow(r9927868, r9927872);
        double r9927874 = r9927871 * r9927873;
        double r9927875 = r9927870 - r9927874;
        double r9927876 = r9927867 * r9927875;
        double r9927877 = -0.3333333333333333;
        double r9927878 = r9927868 * r9927868;
        double r9927879 = r9927868 * r9927878;
        double r9927880 = r9927877 * r9927879;
        double r9927881 = r9927880 * r9927867;
        double r9927882 = r9927876 + r9927881;
        return r9927882;
}

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
Target0.2
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

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

  5. Applied sub-neg0.7

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

  6. Applied associate--l+0.7

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

  7. Applied distribute-lft-in0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019192 
(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))))