Average Error: 43.3 → 0.7
Time: 41.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot im, \frac{-1}{60}, \frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) - \left(im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot im, \frac{-1}{60}, \frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) - \left(im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r6198497 = 0.5;
        double r6198498 = re;
        double r6198499 = sin(r6198498);
        double r6198500 = r6198497 * r6198499;
        double r6198501 = im;
        double r6198502 = -r6198501;
        double r6198503 = exp(r6198502);
        double r6198504 = exp(r6198501);
        double r6198505 = r6198503 - r6198504;
        double r6198506 = r6198500 * r6198505;
        return r6198506;
}
↓
double f(double re, double im) {
        double r6198507 = im;
        double r6198508 = r6198507 * r6198507;
        double r6198509 = r6198508 * r6198508;
        double r6198510 = r6198509 * r6198507;
        double r6198511 = -0.016666666666666666;
        double r6198512 = -0.3333333333333333;
        double r6198513 = r6198507 * r6198508;
        double r6198514 = r6198512 * r6198513;
        double r6198515 = fma(r6198510, r6198511, r6198514);
        double r6198516 = r6198507 + r6198507;
        double r6198517 = r6198515 - r6198516;
        double r6198518 = 0.5;
        double r6198519 = re;
        double r6198520 = sin(r6198519);
        double r6198521 = r6198518 * r6198520;
        double r6198522 = r6198517 * r6198521;
        return r6198522;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.3
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 43.3

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

  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. Using strategy rm

  5. Applied fma-udef0.7

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

  6. Applied associate--r+0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019146 +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))))