Average Error: 58.0 → 0.7
Time: 44.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r6907537 = 0.5;
        double r6907538 = re;
        double r6907539 = cos(r6907538);
        double r6907540 = r6907537 * r6907539;
        double r6907541 = 0.0;
        double r6907542 = im;
        double r6907543 = r6907541 - r6907542;
        double r6907544 = exp(r6907543);
        double r6907545 = exp(r6907542);
        double r6907546 = r6907544 - r6907545;
        double r6907547 = r6907540 * r6907546;
        return r6907547;
}
↓
double f(double re, double im) {
        double r6907548 = -0.3333333333333333;
        double r6907549 = im;
        double r6907550 = r6907549 * r6907549;
        double r6907551 = r6907549 * r6907550;
        double r6907552 = r6907548 * r6907551;
        double r6907553 = 5.0;
        double r6907554 = pow(r6907549, r6907553);
        double r6907555 = -0.016666666666666666;
        double r6907556 = r6907554 * r6907555;
        double r6907557 = r6907549 + r6907549;
        double r6907558 = r6907556 - r6907557;
        double r6907559 = r6907552 + r6907558;
        double r6907560 = 0.5;
        double r6907561 = re;
        double r6907562 = cos(r6907561);
        double r6907563 = r6907560 * r6907562;
        double r6907564 = r6907559 * r6907563;
        return r6907564;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019142 
(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))))