Average Error: 58.0 → 0.8
Time: 28.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot {im}^{3} - \left(im + \left({im}^{5} \cdot \frac{1}{60} + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot {im}^{3} - \left(im + \left({im}^{5} \cdot \frac{1}{60} + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r118645 = 0.5;
        double r118646 = re;
        double r118647 = cos(r118646);
        double r118648 = r118645 * r118647;
        double r118649 = 0.0;
        double r118650 = im;
        double r118651 = r118649 - r118650;
        double r118652 = exp(r118651);
        double r118653 = exp(r118650);
        double r118654 = r118652 - r118653;
        double r118655 = r118648 * r118654;
        return r118655;
}
↓
double f(double re, double im) {
        double r118656 = -0.3333333333333333;
        double r118657 = im;
        double r118658 = 3.0;
        double r118659 = pow(r118657, r118658);
        double r118660 = r118656 * r118659;
        double r118661 = 5.0;
        double r118662 = pow(r118657, r118661);
        double r118663 = 0.016666666666666666;
        double r118664 = r118662 * r118663;
        double r118665 = r118664 + r118657;
        double r118666 = r118657 + r118665;
        double r118667 = r118660 - r118666;
        double r118668 = 0.5;
        double r118669 = re;
        double r118670 = cos(r118669);
        double r118671 = r118668 * r118670;
        double r118672 = r118667 * r118671;
        return r118672;
}

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
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.8

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

  3. Simplified0.8

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

  5. Applied associate--l-0.8

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

  6. Simplified0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019194 
(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))))