Average Error: 57.9 → 0.8
Time: 10.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r132609 = 0.5;
        double r132610 = re;
        double r132611 = cos(r132610);
        double r132612 = r132609 * r132611;
        double r132613 = 0.0;
        double r132614 = im;
        double r132615 = r132613 - r132614;
        double r132616 = exp(r132615);
        double r132617 = exp(r132614);
        double r132618 = r132616 - r132617;
        double r132619 = r132612 * r132618;
        return r132619;
}
↓
double f(double re, double im) {
        double r132620 = 0.5;
        double r132621 = re;
        double r132622 = cos(r132621);
        double r132623 = r132620 * r132622;
        double r132624 = 0.3333333333333333;
        double r132625 = im;
        double r132626 = 3.0;
        double r132627 = pow(r132625, r132626);
        double r132628 = r132624 * r132627;
        double r132629 = 0.016666666666666666;
        double r132630 = 5.0;
        double r132631 = pow(r132625, r132630);
        double r132632 = r132629 * r132631;
        double r132633 = 2.0;
        double r132634 = r132633 * r132625;
        double r132635 = r132632 + r132634;
        double r132636 = r132628 + r132635;
        double r132637 = -r132636;
        double r132638 = r132623 * r132637;
        return r132638;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.9
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2020018 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))