Average Error: 58.0 → 0.7
Time: 36.4s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r9241829 = 0.5;
        double r9241830 = re;
        double r9241831 = cos(r9241830);
        double r9241832 = r9241829 * r9241831;
        double r9241833 = 0.0;
        double r9241834 = im;
        double r9241835 = r9241833 - r9241834;
        double r9241836 = exp(r9241835);
        double r9241837 = exp(r9241834);
        double r9241838 = r9241836 - r9241837;
        double r9241839 = r9241832 * r9241838;
        return r9241839;
}
↓
double f(double re, double im) {
        double r9241840 = -0.3333333333333333;
        double r9241841 = im;
        double r9241842 = r9241841 * r9241841;
        double r9241843 = r9241841 * r9241842;
        double r9241844 = r9241840 * r9241843;
        double r9241845 = r9241841 + r9241841;
        double r9241846 = r9241844 - r9241845;
        double r9241847 = 0.016666666666666666;
        double r9241848 = 5.0;
        double r9241849 = pow(r9241841, r9241848);
        double r9241850 = r9241847 * r9241849;
        double r9241851 = r9241846 - r9241850;
        double r9241852 = 0.5;
        double r9241853 = re;
        double r9241854 = cos(r9241853);
        double r9241855 = r9241852 * r9241854;
        double r9241856 = r9241851 * r9241855;
        return r9241856;
}

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

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

  4. Final simplification0.7

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

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))))