Average Error: 58.2 → 0.6
Time: 42.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\cos re \cdot \left(\left(\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot 0.5\right) + \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(\cos re \cdot 0.5\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\cos re \cdot \left(\left(\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot 0.5\right) + \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(\cos re \cdot 0.5\right)
double f(double re, double im) {
        double r4028420 = 0.5;
        double r4028421 = re;
        double r4028422 = cos(r4028421);
        double r4028423 = r4028420 * r4028422;
        double r4028424 = 0.0;
        double r4028425 = im;
        double r4028426 = r4028424 - r4028425;
        double r4028427 = exp(r4028426);
        double r4028428 = exp(r4028425);
        double r4028429 = r4028427 - r4028428;
        double r4028430 = r4028423 * r4028429;
        return r4028430;
}
↓
double f(double re, double im) {
        double r4028431 = re;
        double r4028432 = cos(r4028431);
        double r4028433 = im;
        double r4028434 = r4028433 * r4028433;
        double r4028435 = r4028434 * r4028434;
        double r4028436 = r4028433 * r4028435;
        double r4028437 = -0.016666666666666666;
        double r4028438 = r4028436 * r4028437;
        double r4028439 = r4028433 + r4028433;
        double r4028440 = r4028438 - r4028439;
        double r4028441 = 0.5;
        double r4028442 = r4028440 * r4028441;
        double r4028443 = r4028432 * r4028442;
        double r4028444 = r4028433 * r4028434;
        double r4028445 = -0.3333333333333333;
        double r4028446 = r4028444 * r4028445;
        double r4028447 = r4028432 * r4028441;
        double r4028448 = r4028446 * r4028447;
        double r4028449 = r4028443 + r4028448;
        return r4028449;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.2

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  5. Applied distribute-rgt-in0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019146 
(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))))