Average Error: 58.0 → 0.8
Time: 38.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right) \cdot im - im \cdot 2\right) - \frac{1}{60} \cdot {im}^{5}\right)\]
\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(\left(im \cdot im\right) \cdot \frac{-1}{3}\right) \cdot im - im \cdot 2\right) - \frac{1}{60} \cdot {im}^{5}\right)
double f(double re, double im) {
        double r6730574 = 0.5;
        double r6730575 = re;
        double r6730576 = cos(r6730575);
        double r6730577 = r6730574 * r6730576;
        double r6730578 = 0.0;
        double r6730579 = im;
        double r6730580 = r6730578 - r6730579;
        double r6730581 = exp(r6730580);
        double r6730582 = exp(r6730579);
        double r6730583 = r6730581 - r6730582;
        double r6730584 = r6730577 * r6730583;
        return r6730584;
}
↓
double f(double re, double im) {
        double r6730585 = 0.5;
        double r6730586 = re;
        double r6730587 = cos(r6730586);
        double r6730588 = r6730585 * r6730587;
        double r6730589 = im;
        double r6730590 = r6730589 * r6730589;
        double r6730591 = -0.3333333333333333;
        double r6730592 = r6730590 * r6730591;
        double r6730593 = r6730592 * r6730589;
        double r6730594 = 2.0;
        double r6730595 = r6730589 * r6730594;
        double r6730596 = r6730593 - r6730595;
        double r6730597 = 0.016666666666666666;
        double r6730598 = 5.0;
        double r6730599 = pow(r6730589, r6730598);
        double r6730600 = r6730597 * r6730599;
        double r6730601 = r6730596 - r6730600;
        double r6730602 = r6730588 * r6730601;
        return r6730602;
}

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

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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019130 
(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))))