Average Error: 57.7 → 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 r362768 = 0.5;
        double r362769 = re;
        double r362770 = cos(r362769);
        double r362771 = r362768 * r362770;
        double r362772 = 0.0;
        double r362773 = im;
        double r362774 = r362772 - r362773;
        double r362775 = exp(r362774);
        double r362776 = exp(r362773);
        double r362777 = r362775 - r362776;
        double r362778 = r362771 * r362777;
        return r362778;
}
↓
double f(double re, double im) {
        double r362779 = 0.5;
        double r362780 = re;
        double r362781 = cos(r362780);
        double r362782 = r362779 * r362781;
        double r362783 = 0.3333333333333333;
        double r362784 = im;
        double r362785 = 3.0;
        double r362786 = pow(r362784, r362785);
        double r362787 = r362783 * r362786;
        double r362788 = 0.016666666666666666;
        double r362789 = 5.0;
        double r362790 = pow(r362784, r362789);
        double r362791 = r362788 * r362790;
        double r362792 = 2.0;
        double r362793 = r362792 * r362784;
        double r362794 = r362791 + r362793;
        double r362795 = r362787 + r362794;
        double r362796 = -r362795;
        double r362797 = r362782 * r362796;
        return r362797;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.7
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 57.7

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

  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 2020100 
(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))))