Average Error: 43.3 → 0.8
Time: 31.0s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r9064346 = 0.5;
        double r9064347 = re;
        double r9064348 = sin(r9064347);
        double r9064349 = r9064346 * r9064348;
        double r9064350 = im;
        double r9064351 = -r9064350;
        double r9064352 = exp(r9064351);
        double r9064353 = exp(r9064350);
        double r9064354 = r9064352 - r9064353;
        double r9064355 = r9064349 * r9064354;
        return r9064355;
}
↓
double f(double re, double im) {
        double r9064356 = -0.3333333333333333;
        double r9064357 = im;
        double r9064358 = r9064357 * r9064357;
        double r9064359 = r9064357 * r9064358;
        double r9064360 = r9064356 * r9064359;
        double r9064361 = 5.0;
        double r9064362 = pow(r9064357, r9064361);
        double r9064363 = 0.016666666666666666;
        double r9064364 = r9064357 + r9064357;
        double r9064365 = fma(r9064362, r9064363, r9064364);
        double r9064366 = r9064360 - r9064365;
        double r9064367 = 0.5;
        double r9064368 = re;
        double r9064369 = sin(r9064368);
        double r9064370 = r9064367 * r9064369;
        double r9064371 = r9064366 * r9064370;
        return r9064371;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.3
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.3

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(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))))