Average Error: 43.6 → 0.8
Time: 18.2s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{3}, {im}^{3}, \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{3}, {im}^{3}, \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r365503 = 0.5;
        double r365504 = re;
        double r365505 = sin(r365504);
        double r365506 = r365503 * r365505;
        double r365507 = im;
        double r365508 = -r365507;
        double r365509 = exp(r365508);
        double r365510 = exp(r365507);
        double r365511 = r365509 - r365510;
        double r365512 = r365506 * r365511;
        return r365512;
}
↓
double f(double re, double im) {
        double r365513 = 0.5;
        double r365514 = re;
        double r365515 = sin(r365514);
        double r365516 = r365513 * r365515;
        double r365517 = 0.3333333333333333;
        double r365518 = im;
        double r365519 = 3.0;
        double r365520 = pow(r365518, r365519);
        double r365521 = 0.016666666666666666;
        double r365522 = 5.0;
        double r365523 = pow(r365518, r365522);
        double r365524 = 2.0;
        double r365525 = r365524 * r365518;
        double r365526 = fma(r365521, r365523, r365525);
        double r365527 = fma(r365517, r365520, r365526);
        double r365528 = -r365527;
        double r365529 = r365516 * r365528;
        return r365529;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.6
Target0.2
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 43.6

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

  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. Simplified0.8

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

  4. Final simplification0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{3}, {im}^{3}, \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))