Average Error: 58.0 → 0.8
Time: 10.3s
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 r623439 = 0.5;
        double r623440 = re;
        double r623441 = cos(r623440);
        double r623442 = r623439 * r623441;
        double r623443 = 0.0;
        double r623444 = im;
        double r623445 = r623443 - r623444;
        double r623446 = exp(r623445);
        double r623447 = exp(r623444);
        double r623448 = r623446 - r623447;
        double r623449 = r623442 * r623448;
        return r623449;
}
↓
double f(double re, double im) {
        double r623450 = 0.5;
        double r623451 = re;
        double r623452 = cos(r623451);
        double r623453 = r623450 * r623452;
        double r623454 = 0.3333333333333333;
        double r623455 = im;
        double r623456 = 3.0;
        double r623457 = pow(r623455, r623456);
        double r623458 = r623454 * r623457;
        double r623459 = 0.016666666666666666;
        double r623460 = 5.0;
        double r623461 = pow(r623455, r623460);
        double r623462 = r623459 * r623461;
        double r623463 = 2.0;
        double r623464 = r623463 * r623455;
        double r623465 = r623462 + r623464;
        double r623466 = r623458 + r623465;
        double r623467 = -r623466;
        double r623468 = r623453 * r623467;
        return r623468;
}

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

Derivation

  1. Initial program 58.0

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

  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 2020039 
(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))))