Average Error: 43.4 → 0.7
Time: 37.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) + \frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) + \frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r11299534 = 0.5;
        double r11299535 = re;
        double r11299536 = sin(r11299535);
        double r11299537 = r11299534 * r11299536;
        double r11299538 = im;
        double r11299539 = -r11299538;
        double r11299540 = exp(r11299539);
        double r11299541 = exp(r11299538);
        double r11299542 = r11299540 - r11299541;
        double r11299543 = r11299537 * r11299542;
        return r11299543;
}
↓
double f(double re, double im) {
        double r11299544 = im;
        double r11299545 = 5.0;
        double r11299546 = pow(r11299544, r11299545);
        double r11299547 = -0.016666666666666666;
        double r11299548 = r11299546 * r11299547;
        double r11299549 = r11299544 + r11299544;
        double r11299550 = r11299548 - r11299549;
        double r11299551 = -0.3333333333333333;
        double r11299552 = r11299544 * r11299544;
        double r11299553 = r11299544 * r11299552;
        double r11299554 = r11299551 * r11299553;
        double r11299555 = r11299550 + r11299554;
        double r11299556 = 0.5;
        double r11299557 = re;
        double r11299558 = sin(r11299557);
        double r11299559 = r11299556 * r11299558;
        double r11299560 = r11299555 * r11299559;
        return r11299560;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019169 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))