Average Error: 43.2 → 0.7
Time: 43.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\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(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r8349346 = 0.5;
        double r8349347 = re;
        double r8349348 = sin(r8349347);
        double r8349349 = r8349346 * r8349348;
        double r8349350 = im;
        double r8349351 = -r8349350;
        double r8349352 = exp(r8349351);
        double r8349353 = exp(r8349350);
        double r8349354 = r8349352 - r8349353;
        double r8349355 = r8349349 * r8349354;
        return r8349355;
}
↓
double f(double re, double im) {
        double r8349356 = -0.3333333333333333;
        double r8349357 = im;
        double r8349358 = r8349357 * r8349357;
        double r8349359 = r8349357 * r8349358;
        double r8349360 = r8349356 * r8349359;
        double r8349361 = 0.016666666666666666;
        double r8349362 = 5.0;
        double r8349363 = pow(r8349357, r8349362);
        double r8349364 = r8349357 + r8349357;
        double r8349365 = fma(r8349361, r8349363, r8349364);
        double r8349366 = r8349360 - r8349365;
        double r8349367 = 0.5;
        double r8349368 = re;
        double r8349369 = sin(r8349368);
        double r8349370 = r8349367 * r8349369;
        double r8349371 = r8349366 * r8349370;
        return r8349371;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.2
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 43.2

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 +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))))