Average Error: 58.1 → 0.7
Time: 38.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[-\left(0.1666666666666666574148081281236954964697 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.008333333333333333217685101601546193705872 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
-\left(0.1666666666666666574148081281236954964697 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.008333333333333333217685101601546193705872 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)
double f(double re, double im) {
        double r156776 = 0.5;
        double r156777 = re;
        double r156778 = cos(r156777);
        double r156779 = r156776 * r156778;
        double r156780 = 0.0;
        double r156781 = im;
        double r156782 = r156780 - r156781;
        double r156783 = exp(r156782);
        double r156784 = exp(r156781);
        double r156785 = r156783 - r156784;
        double r156786 = r156779 * r156785;
        return r156786;
}
↓
double f(double re, double im) {
        double r156787 = 0.16666666666666666;
        double r156788 = re;
        double r156789 = cos(r156788);
        double r156790 = im;
        double r156791 = 3.0;
        double r156792 = pow(r156790, r156791);
        double r156793 = r156789 * r156792;
        double r156794 = r156787 * r156793;
        double r156795 = 0.008333333333333333;
        double r156796 = 5.0;
        double r156797 = pow(r156790, r156796);
        double r156798 = r156789 * r156797;
        double r156799 = r156795 * r156798;
        double r156800 = 1.0;
        double r156801 = r156789 * r156790;
        double r156802 = r156800 * r156801;
        double r156803 = r156799 + r156802;
        double r156804 = r156794 + r156803;
        double r156805 = -r156804;
        return r156805;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
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 58.1

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

  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. Taylor expanded around inf 0.7

    \[\leadsto \color{blue}{-\left(0.1666666666666666574148081281236954964697 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.008333333333333333217685101601546193705872 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto -\left(0.1666666666666666574148081281236954964697 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.008333333333333333217685101601546193705872 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(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))))