Average Error: 43.4 → 0.8
Time: 31.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.16666666666666666 \cdot \sin re\right) \cdot \left(-im \cdot \left(im \cdot im\right)\right) - \left(\left(im \cdot \sin re\right) \cdot 1.0 + \left(0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.16666666666666666 \cdot \sin re\right) \cdot \left(-im \cdot \left(im \cdot im\right)\right) - \left(\left(im \cdot \sin re\right) \cdot 1.0 + \left(0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re\right)
double f(double re, double im) {
        double r7298638 = 0.5;
        double r7298639 = re;
        double r7298640 = sin(r7298639);
        double r7298641 = r7298638 * r7298640;
        double r7298642 = im;
        double r7298643 = -r7298642;
        double r7298644 = exp(r7298643);
        double r7298645 = exp(r7298642);
        double r7298646 = r7298644 - r7298645;
        double r7298647 = r7298641 * r7298646;
        return r7298647;
}
↓
double f(double re, double im) {
        double r7298648 = 0.16666666666666666;
        double r7298649 = re;
        double r7298650 = sin(r7298649);
        double r7298651 = r7298648 * r7298650;
        double r7298652 = im;
        double r7298653 = r7298652 * r7298652;
        double r7298654 = r7298652 * r7298653;
        double r7298655 = -r7298654;
        double r7298656 = r7298651 * r7298655;
        double r7298657 = r7298652 * r7298650;
        double r7298658 = 1.0;
        double r7298659 = r7298657 * r7298658;
        double r7298660 = 0.008333333333333333;
        double r7298661 = 5.0;
        double r7298662 = pow(r7298652, r7298661);
        double r7298663 = r7298660 * r7298662;
        double r7298664 = r7298663 * r7298650;
        double r7298665 = r7298659 + r7298664;
        double r7298666 = r7298656 - r7298665;
        return r7298666;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Taylor expanded around inf 0.8

    \[\leadsto \color{blue}{-\left(0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1.0 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

  5. Simplified0.8

    \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \sin re\right) \cdot \left(-im \cdot \left(im \cdot im\right)\right) - \left(\left(\sin re \cdot im\right) \cdot 1.0 + \left(0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re\right)}\]

  6. Final simplification0.8

    \[\leadsto \left(0.16666666666666666 \cdot \sin re\right) \cdot \left(-im \cdot \left(im \cdot im\right)\right) - \left(\left(im \cdot \sin re\right) \cdot 1.0 + \left(0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re\right)\]

Reproduce

herbie shell --seed 2019134 
(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))))