math.cos on complex, imaginary part

Average Error: 43.3 → 0.8

Time: 8.9s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r234642 = 0.5;
        double r234643 = re;
        double r234644 = sin(r234643);
        double r234645 = r234642 * r234644;
        double r234646 = im;
        double r234647 = -r234646;
        double r234648 = exp(r234647);
        double r234649 = exp(r234646);
        double r234650 = r234648 - r234649;
        double r234651 = r234645 * r234650;
        return r234651;
}

↓

double f(double re, double im) {
        double r234652 = 0.5;
        double r234653 = re;
        double r234654 = sin(r234653);
        double r234655 = -0.3333333333333333;
        double r234656 = im;
        double r234657 = 3.0;
        double r234658 = pow(r234656, r234657);
        double r234659 = r234655 * r234658;
        double r234660 = r234654 * r234659;
        double r234661 = r234652 * r234660;
        double r234662 = r234652 * r234654;
        double r234663 = 0.016666666666666666;
        double r234664 = 5.0;
        double r234665 = pow(r234656, r234664);
        double r234666 = 2.0;
        double r234667 = r234666 * r234656;
        double r234668 = fma(r234663, r234665, r234667);
        double r234669 = -r234668;
        double r234670 = r234662 * r234669;
        double r234671 = r234661 + r234670;
        return r234671;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.3
Target	0.3
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.8

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

3. Simplified 0.8

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

5. Applied sub-neg 0.8

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

6. Applied distribute-lft-in 0.8

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

7. Simplified 0.8

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

8. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))