math.cos on complex, imaginary part

Average Error: 43.2 → 0.8

Time: 21.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double re, double im) {
        double r251714 = 0.5;
        double r251715 = re;
        double r251716 = sin(r251715);
        double r251717 = r251714 * r251716;
        double r251718 = im;
        double r251719 = -r251718;
        double r251720 = exp(r251719);
        double r251721 = exp(r251718);
        double r251722 = r251720 - r251721;
        double r251723 = r251717 * r251722;
        return r251723;
}

↓

double f(double re, double im) {
        double r251724 = 0.5;
        double r251725 = re;
        double r251726 = sin(r251725);
        double r251727 = r251724 * r251726;
        double r251728 = im;
        double r251729 = 3.0;
        double r251730 = pow(r251728, r251729);
        double r251731 = -0.3333333333333333;
        double r251732 = r251730 * r251731;
        double r251733 = 0.016666666666666666;
        double r251734 = cbrt(r251728);
        double r251735 = r251734 * r251734;
        double r251736 = 5.0;
        double r251737 = pow(r251735, r251736);
        double r251738 = r251733 * r251737;
        double r251739 = cbrt(r251734);
        double r251740 = r251739 * r251739;
        double r251741 = pow(r251740, r251736);
        double r251742 = pow(r251739, r251736);
        double r251743 = r251741 * r251742;
        double r251744 = r251738 * r251743;
        double r251745 = 2.0;
        double r251746 = r251745 * r251728;
        double r251747 = r251744 + r251746;
        double r251748 = r251732 - r251747;
        double r251749 = r251727 * r251748;
        return r251749;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.2
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 43.2

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

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

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

5. Applied add-cube-cbrt 0.8

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

6. Applied unpow-prod-down 0.8

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

7. Applied associate-*r* 0.8

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

9. Applied add-cube-cbrt 0.8

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

10. Applied unpow-prod-down 0.8

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

11. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020045 
(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))))