Average Error: 58.0 → 0.8
Time: 39.4s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - 2 \cdot im\right) - \frac{1}{60} \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - 2 \cdot im\right) - \frac{1}{60} \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)
double f(double re, double im) {
        double r8628690 = 0.5;
        double r8628691 = re;
        double r8628692 = cos(r8628691);
        double r8628693 = r8628690 * r8628692;
        double r8628694 = 0.0;
        double r8628695 = im;
        double r8628696 = r8628694 - r8628695;
        double r8628697 = exp(r8628696);
        double r8628698 = exp(r8628695);
        double r8628699 = r8628697 - r8628698;
        double r8628700 = r8628693 * r8628699;
        return r8628700;
}
↓
double f(double re, double im) {
        double r8628701 = 0.5;
        double r8628702 = re;
        double r8628703 = cos(r8628702);
        double r8628704 = r8628701 * r8628703;
        double r8628705 = im;
        double r8628706 = r8628705 * r8628705;
        double r8628707 = r8628705 * r8628706;
        double r8628708 = -0.3333333333333333;
        double r8628709 = r8628707 * r8628708;
        double r8628710 = 2.0;
        double r8628711 = r8628710 * r8628705;
        double r8628712 = r8628709 - r8628711;
        double r8628713 = 0.016666666666666666;
        double r8628714 = r8628706 * r8628706;
        double r8628715 = r8628705 * r8628714;
        double r8628716 = r8628713 * r8628715;
        double r8628717 = r8628712 - r8628716;
        double r8628718 = r8628704 * r8628717;
        return r8628718;
}

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
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  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

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

  4. Taylor expanded around inf 0.8

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

  5. Simplified0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019138 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))