Average Error: 43.5 → 0.7
Time: 33.2s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\left(im + {im}^{5} \cdot \frac{1}{60}\right) + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\left(im + {im}^{5} \cdot \frac{1}{60}\right) + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r12578597 = 0.5;
        double r12578598 = re;
        double r12578599 = sin(r12578598);
        double r12578600 = r12578597 * r12578599;
        double r12578601 = im;
        double r12578602 = -r12578601;
        double r12578603 = exp(r12578602);
        double r12578604 = exp(r12578601);
        double r12578605 = r12578603 - r12578604;
        double r12578606 = r12578600 * r12578605;
        return r12578606;
}
↓
double f(double re, double im) {
        double r12578607 = im;
        double r12578608 = r12578607 * r12578607;
        double r12578609 = r12578607 * r12578608;
        double r12578610 = -0.3333333333333333;
        double r12578611 = r12578609 * r12578610;
        double r12578612 = 5.0;
        double r12578613 = pow(r12578607, r12578612);
        double r12578614 = 0.016666666666666666;
        double r12578615 = r12578613 * r12578614;
        double r12578616 = r12578607 + r12578615;
        double r12578617 = r12578616 + r12578607;
        double r12578618 = r12578611 - r12578617;
        double r12578619 = 0.5;
        double r12578620 = re;
        double r12578621 = sin(r12578620);
        double r12578622 = r12578619 * r12578621;
        double r12578623 = r12578618 * r12578622;
        return r12578623;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.5
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019164 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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