Average Error: 43.8 → 0.7
Time: 34.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\frac{-1}{3} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\frac{-1}{3} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r9814533 = 0.5;
        double r9814534 = re;
        double r9814535 = sin(r9814534);
        double r9814536 = r9814533 * r9814535;
        double r9814537 = im;
        double r9814538 = -r9814537;
        double r9814539 = exp(r9814538);
        double r9814540 = exp(r9814537);
        double r9814541 = r9814539 - r9814540;
        double r9814542 = r9814536 * r9814541;
        return r9814542;
}
↓
double f(double re, double im) {
        double r9814543 = im;
        double r9814544 = 5.0;
        double r9814545 = pow(r9814543, r9814544);
        double r9814546 = -0.016666666666666666;
        double r9814547 = r9814545 * r9814546;
        double r9814548 = r9814543 + r9814543;
        double r9814549 = r9814547 - r9814548;
        double r9814550 = 0.5;
        double r9814551 = re;
        double r9814552 = sin(r9814551);
        double r9814553 = r9814550 * r9814552;
        double r9814554 = r9814549 * r9814553;
        double r9814555 = -0.3333333333333333;
        double r9814556 = r9814543 * r9814543;
        double r9814557 = r9814556 * r9814543;
        double r9814558 = r9814555 * r9814557;
        double r9814559 = r9814558 * r9814553;
        double r9814560 = r9814554 + r9814559;
        return r9814560;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.8
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.8

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

  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

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

  5. Applied distribute-lft-in0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019170 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))