math.cos on complex, imaginary part

Average Error: 43.2 → 0.8

Time: 18.9s

Precision: 64

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

Math

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

↓

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

C

double f(double re, double im) {
        double r329810 = 0.5;
        double r329811 = re;
        double r329812 = sin(r329811);
        double r329813 = r329810 * r329812;
        double r329814 = im;
        double r329815 = -r329814;
        double r329816 = exp(r329815);
        double r329817 = exp(r329814);
        double r329818 = r329816 - r329817;
        double r329819 = r329813 * r329818;
        return r329819;
}

↓

double f(double re, double im) {
        double r329820 = 0.5;
        double r329821 = re;
        double r329822 = sin(r329821);
        double r329823 = r329820 * r329822;
        double r329824 = -0.3333333333333333;
        double r329825 = im;
        double r329826 = 3.0;
        double r329827 = pow(r329825, r329826);
        double r329828 = r329824 * r329827;
        double r329829 = 0.016666666666666666;
        double r329830 = cbrt(r329825);
        double r329831 = r329830 * r329830;
        double r329832 = 5.0;
        double r329833 = pow(r329831, r329832);
        double r329834 = r329829 * r329833;
        double r329835 = cbrt(r329830);
        double r329836 = r329835 * r329835;
        double r329837 = pow(r329836, r329832);
        double r329838 = pow(r329835, r329832);
        double r329839 = r329837 * r329838;
        double r329840 = r329834 * r329839;
        double r329841 = 2.0;
        double r329842 = r329841 * r329825;
        double r329843 = r329840 + r329842;
        double r329844 = r329828 - r329843;
        double r329845 = r329823 * r329844;
        return r329845;
}

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

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

5. Applied add-cube-cbrt 0.8

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

6. Applied unpow-prod-down 0.8

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

7. Applied associate-*r* 0.8

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

9. Applied add-cube-cbrt 0.8

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

10. Applied unpow-prod-down 0.8

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

11. Final simplification 0.8

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

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))))