Average Error: 58.1 → 0.7
Time: 28.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{3}, {im}^{3}, \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos 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 r161408 = 0.5;
        double r161409 = re;
        double r161410 = cos(r161409);
        double r161411 = r161408 * r161410;
        double r161412 = 0.0;
        double r161413 = im;
        double r161414 = r161412 - r161413;
        double r161415 = exp(r161414);
        double r161416 = exp(r161413);
        double r161417 = r161415 - r161416;
        double r161418 = r161411 * r161417;
        return r161418;
}
↓
double f(double re, double im) {
        double r161419 = 0.5;
        double r161420 = re;
        double r161421 = cos(r161420);
        double r161422 = r161419 * r161421;
        double r161423 = 0.3333333333333333;
        double r161424 = im;
        double r161425 = 3.0;
        double r161426 = pow(r161424, r161425);
        double r161427 = 0.016666666666666666;
        double r161428 = 5.0;
        double r161429 = pow(r161424, r161428);
        double r161430 = 2.0;
        double r161431 = r161430 * r161424;
        double r161432 = fma(r161427, r161429, r161431);
        double r161433 = fma(r161423, r161426, r161432);
        double r161434 = -r161433;
        double r161435 = r161422 * r161434;
        return r161435;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.1
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))