math.sin on complex, imaginary part

Average Error: 58.2 → 0.6

Time: 1.9m

Precision: 64

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

↓

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

C

double f(double re, double im) {
        double r46113847 = 0.5;
        double r46113848 = re;
        double r46113849 = cos(r46113848);
        double r46113850 = r46113847 * r46113849;
        double r46113851 = 0.0;
        double r46113852 = im;
        double r46113853 = r46113851 - r46113852;
        double r46113854 = exp(r46113853);
        double r46113855 = exp(r46113852);
        double r46113856 = r46113854 - r46113855;
        double r46113857 = r46113850 * r46113856;
        return r46113857;
}

↓

double f(double re, double im) {
        double r46113858 = im;
        double r46113859 = 5.0;
        double r46113860 = pow(r46113858, r46113859);
        double r46113861 = -0.016666666666666666;
        double r46113862 = -2.0;
        double r46113863 = r46113858 * r46113862;
        double r46113864 = -0.3333333333333333;
        double r46113865 = r46113858 * r46113864;
        double r46113866 = r46113858 * r46113865;
        double r46113867 = r46113858 * r46113866;
        double r46113868 = r46113863 + r46113867;
        double r46113869 = fma(r46113860, r46113861, r46113868);
        double r46113870 = 0.5;
        double r46113871 = re;
        double r46113872 = cos(r46113871);
        double r46113873 = r46113870 * r46113872;
        double r46113874 = r46113869 * r46113873;
        return r46113874;
}

Error

Bits error versus re

Bits error versus im

Target

Original	58.2
Target	0.2
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 58.2

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

2. Taylor expanded around 0 0.6

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

3. Simplified 0.6

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

5. Applied sub-neg 0.6

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

6. Applied distribute-lft-in 0.6

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

7. Simplified 0.6

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

8. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))