Average Error: 58.2 → 0.7
Time: 29.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\left({im}^{5} \cdot \frac{-1}{60} - \left(2 - \left(im \cdot \frac{-1}{3}\right) \cdot im\right) \cdot im\right) \cdot \cos re\right) \cdot 0.5\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\left({im}^{5} \cdot \frac{-1}{60} - \left(2 - \left(im \cdot \frac{-1}{3}\right) \cdot im\right) \cdot im\right) \cdot \cos re\right) \cdot 0.5
double f(double re, double im) {
        double r6694830 = 0.5;
        double r6694831 = re;
        double r6694832 = cos(r6694831);
        double r6694833 = r6694830 * r6694832;
        double r6694834 = 0.0;
        double r6694835 = im;
        double r6694836 = r6694834 - r6694835;
        double r6694837 = exp(r6694836);
        double r6694838 = exp(r6694835);
        double r6694839 = r6694837 - r6694838;
        double r6694840 = r6694833 * r6694839;
        return r6694840;
}
↓
double f(double re, double im) {
        double r6694841 = im;
        double r6694842 = 5.0;
        double r6694843 = pow(r6694841, r6694842);
        double r6694844 = -0.016666666666666666;
        double r6694845 = r6694843 * r6694844;
        double r6694846 = 2.0;
        double r6694847 = -0.3333333333333333;
        double r6694848 = r6694841 * r6694847;
        double r6694849 = r6694848 * r6694841;
        double r6694850 = r6694846 - r6694849;
        double r6694851 = r6694850 * r6694841;
        double r6694852 = r6694845 - r6694851;
        double r6694853 = re;
        double r6694854 = cos(r6694853);
        double r6694855 = r6694852 * r6694854;
        double r6694856 = 0.5;
        double r6694857 = r6694855 * r6694856;
        return r6694857;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
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 58.2

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

  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

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

  5. Applied associate-*l*0.7

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

  6. Simplified0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 
(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))))