Average Error: 43.7 → 0.8
Time: 7.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[-\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
-\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)
double f(double re, double im) {
        double r395347 = 0.5;
        double r395348 = re;
        double r395349 = sin(r395348);
        double r395350 = r395347 * r395349;
        double r395351 = im;
        double r395352 = -r395351;
        double r395353 = exp(r395352);
        double r395354 = exp(r395351);
        double r395355 = r395353 - r395354;
        double r395356 = r395350 * r395355;
        return r395356;
}
↓
double f(double re, double im) {
        double r395357 = 0.16666666666666666;
        double r395358 = re;
        double r395359 = sin(r395358);
        double r395360 = im;
        double r395361 = 3.0;
        double r395362 = pow(r395360, r395361);
        double r395363 = r395359 * r395362;
        double r395364 = r395357 * r395363;
        double r395365 = 1.0;
        double r395366 = r395359 * r395360;
        double r395367 = r395365 * r395366;
        double r395368 = 0.008333333333333333;
        double r395369 = 5.0;
        double r395370 = pow(r395360, r395369);
        double r395371 = r395359 * r395370;
        double r395372 = r395368 * r395371;
        double r395373 = r395367 + r395372;
        double r395374 = r395364 + r395373;
        double r395375 = -r395374;
        return r395375;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.7
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.8

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

  3. Taylor expanded around inf 0.8

    \[\leadsto \color{blue}{-\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

  4. Final simplification0.8

    \[\leadsto -\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020021 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))