Average Error: 58.1 → 0.7
Time: 10.0s
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 r230483 = 0.5;
        double r230484 = re;
        double r230485 = cos(r230484);
        double r230486 = r230483 * r230485;
        double r230487 = 0.0;
        double r230488 = im;
        double r230489 = r230487 - r230488;
        double r230490 = exp(r230489);
        double r230491 = exp(r230488);
        double r230492 = r230490 - r230491;
        double r230493 = r230486 * r230492;
        return r230493;
}
↓
double f(double re, double im) {
        double r230494 = 0.5;
        double r230495 = re;
        double r230496 = cos(r230495);
        double r230497 = r230494 * r230496;
        double r230498 = 0.3333333333333333;
        double r230499 = im;
        double r230500 = 3.0;
        double r230501 = pow(r230499, r230500);
        double r230502 = r230498 * r230501;
        double r230503 = 0.016666666666666666;
        double r230504 = 5.0;
        double r230505 = pow(r230499, r230504);
        double r230506 = r230503 * r230505;
        double r230507 = 2.0;
        double r230508 = r230507 * r230499;
        double r230509 = r230506 + r230508;
        double r230510 = r230502 + r230509;
        double r230511 = -r230510;
        double r230512 = r230497 * r230511;
        return r230512;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

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

  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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019354 
(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))))