Average Error: 58.0 → 0.8
Time: 9.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\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(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r355858 = 0.5;
        double r355859 = re;
        double r355860 = cos(r355859);
        double r355861 = r355858 * r355860;
        double r355862 = 0.0;
        double r355863 = im;
        double r355864 = r355862 - r355863;
        double r355865 = exp(r355864);
        double r355866 = exp(r355863);
        double r355867 = r355865 - r355866;
        double r355868 = r355861 * r355867;
        return r355868;
}
↓
double f(double re, double im) {
        double r355869 = 0.5;
        double r355870 = re;
        double r355871 = cos(r355870);
        double r355872 = r355869 * r355871;
        double r355873 = 0.3333333333333333;
        double r355874 = im;
        double r355875 = 3.0;
        double r355876 = pow(r355874, r355875);
        double r355877 = r355873 * r355876;
        double r355878 = 0.016666666666666666;
        double r355879 = 5.0;
        double r355880 = pow(r355874, r355879);
        double r355881 = r355878 * r355880;
        double r355882 = 2.0;
        double r355883 = r355882 * r355874;
        double r355884 = r355881 + r355883;
        double r355885 = r355877 + r355884;
        double r355886 = -r355885;
        double r355887 = r355872 * r355886;
        return r355887;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020056 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))