Average Error: 43.9 → 0.7
Time: 54.4s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[(\left({im}^{5}\right) \cdot \frac{-1}{60} + \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right))_* \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
(\left({im}^{5}\right) \cdot \frac{-1}{60} + \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right))_* \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r44583638 = 0.5;
        double r44583639 = re;
        double r44583640 = sin(r44583639);
        double r44583641 = r44583638 * r44583640;
        double r44583642 = im;
        double r44583643 = -r44583642;
        double r44583644 = exp(r44583643);
        double r44583645 = exp(r44583642);
        double r44583646 = r44583644 - r44583645;
        double r44583647 = r44583641 * r44583646;
        return r44583647;
}
↓
double f(double re, double im) {
        double r44583648 = im;
        double r44583649 = 5.0;
        double r44583650 = pow(r44583648, r44583649);
        double r44583651 = -0.016666666666666666;
        double r44583652 = -0.3333333333333333;
        double r44583653 = r44583648 * r44583652;
        double r44583654 = r44583648 * r44583653;
        double r44583655 = 2.0;
        double r44583656 = r44583654 - r44583655;
        double r44583657 = r44583648 * r44583656;
        double r44583658 = fma(r44583650, r44583651, r44583657);
        double r44583659 = 0.5;
        double r44583660 = re;
        double r44583661 = sin(r44583660);
        double r44583662 = r44583659 * r44583661;
        double r44583663 = r44583658 * r44583662;
        return r44583663;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.9
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 43.9

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(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))))