Average Error: 58.2 → 0.7
Time: 32.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, \left(im \cdot im\right) \cdot \left(\frac{-1}{3} \cdot im\right)\right) - im \cdot 2\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, \left(im \cdot im\right) \cdot \left(\frac{-1}{3} \cdot im\right)\right) - im \cdot 2\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r5167603 = 0.5;
        double r5167604 = re;
        double r5167605 = cos(r5167604);
        double r5167606 = r5167603 * r5167605;
        double r5167607 = 0.0;
        double r5167608 = im;
        double r5167609 = r5167607 - r5167608;
        double r5167610 = exp(r5167609);
        double r5167611 = exp(r5167608);
        double r5167612 = r5167610 - r5167611;
        double r5167613 = r5167606 * r5167612;
        return r5167613;
}
↓
double f(double re, double im) {
        double r5167614 = -0.016666666666666666;
        double r5167615 = im;
        double r5167616 = 5.0;
        double r5167617 = pow(r5167615, r5167616);
        double r5167618 = r5167615 * r5167615;
        double r5167619 = -0.3333333333333333;
        double r5167620 = r5167619 * r5167615;
        double r5167621 = r5167618 * r5167620;
        double r5167622 = fma(r5167614, r5167617, r5167621);
        double r5167623 = 2.0;
        double r5167624 = r5167615 * r5167623;
        double r5167625 = r5167622 - r5167624;
        double r5167626 = 0.5;
        double r5167627 = re;
        double r5167628 = cos(r5167627);
        double r5167629 = r5167626 * r5167628;
        double r5167630 = r5167625 * r5167629;
        return r5167630;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.2
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 58.2

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

  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

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

  5. Applied fma-udef0.7

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

  6. Applied associate--r+0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 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))))