Average Error: 57.9 → 0.8
Time: 9.9s
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 r223357 = 0.5;
        double r223358 = re;
        double r223359 = cos(r223358);
        double r223360 = r223357 * r223359;
        double r223361 = 0.0;
        double r223362 = im;
        double r223363 = r223361 - r223362;
        double r223364 = exp(r223363);
        double r223365 = exp(r223362);
        double r223366 = r223364 - r223365;
        double r223367 = r223360 * r223366;
        return r223367;
}
↓
double f(double re, double im) {
        double r223368 = 0.5;
        double r223369 = re;
        double r223370 = cos(r223369);
        double r223371 = r223368 * r223370;
        double r223372 = 0.3333333333333333;
        double r223373 = im;
        double r223374 = 3.0;
        double r223375 = pow(r223373, r223374);
        double r223376 = r223372 * r223375;
        double r223377 = 0.016666666666666666;
        double r223378 = 5.0;
        double r223379 = pow(r223373, r223378);
        double r223380 = r223377 * r223379;
        double r223381 = 2.0;
        double r223382 = r223381 * r223373;
        double r223383 = r223380 + r223382;
        double r223384 = r223376 + r223383;
        double r223385 = -r223384;
        double r223386 = r223371 * r223385;
        return r223386;
}

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 2020021 
(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))))