Average Error: 43.7 → 0.7
Time: 28.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{-1}{3}, im, \frac{-1}{60} \cdot {im}^{5}\right) - \left(im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{-1}{3}, im, \frac{-1}{60} \cdot {im}^{5}\right) - \left(im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r6203488 = 0.5;
        double r6203489 = re;
        double r6203490 = sin(r6203489);
        double r6203491 = r6203488 * r6203490;
        double r6203492 = im;
        double r6203493 = -r6203492;
        double r6203494 = exp(r6203493);
        double r6203495 = exp(r6203492);
        double r6203496 = r6203494 - r6203495;
        double r6203497 = r6203491 * r6203496;
        return r6203497;
}
↓
double f(double re, double im) {
        double r6203498 = im;
        double r6203499 = r6203498 * r6203498;
        double r6203500 = -0.3333333333333333;
        double r6203501 = r6203499 * r6203500;
        double r6203502 = -0.016666666666666666;
        double r6203503 = 5.0;
        double r6203504 = pow(r6203498, r6203503);
        double r6203505 = r6203502 * r6203504;
        double r6203506 = fma(r6203501, r6203498, r6203505);
        double r6203507 = r6203498 + r6203498;
        double r6203508 = r6203506 - r6203507;
        double r6203509 = 0.5;
        double r6203510 = re;
        double r6203511 = sin(r6203510);
        double r6203512 = r6203509 * r6203511;
        double r6203513 = r6203508 * r6203512;
        return r6203513;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.7
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  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

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

  5. Applied fma-udef0.7

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

  6. Applied associate--r+0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 +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))))