math.cos on complex, imaginary part

Average Error: 43.5 → 0.8

Time: 8.7s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r297776 = 0.5;
        double r297777 = re;
        double r297778 = sin(r297777);
        double r297779 = r297776 * r297778;
        double r297780 = im;
        double r297781 = -r297780;
        double r297782 = exp(r297781);
        double r297783 = exp(r297780);
        double r297784 = r297782 - r297783;
        double r297785 = r297779 * r297784;
        return r297785;
}

↓

double f(double re, double im) {
        double r297786 = 0.5;
        double r297787 = re;
        double r297788 = sin(r297787);
        double r297789 = r297786 * r297788;
        double r297790 = im;
        double r297791 = 3.0;
        double r297792 = pow(r297790, r297791);
        double r297793 = r297789 * r297792;
        double r297794 = -0.3333333333333333;
        double r297795 = r297793 * r297794;
        double r297796 = 0.016666666666666666;
        double r297797 = 5.0;
        double r297798 = pow(r297790, r297797);
        double r297799 = r297796 * r297798;
        double r297800 = cbrt(r297799);
        double r297801 = r297800 * r297800;
        double r297802 = cbrt(r297801);
        double r297803 = r297802 * r297802;
        double r297804 = r297803 * r297802;
        double r297805 = r297804 * r297800;
        double r297806 = 2.0;
        double r297807 = r297806 * r297790;
        double r297808 = r297805 + r297807;
        double r297809 = -r297808;
        double r297810 = r297789 * r297809;
        double r297811 = r297795 + r297810;
        return r297811;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.5
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 43.5

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

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. Using strategy rm

4. Applied distribute-neg-in 0.8

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

5. Applied distribute-lft-in 0.8

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

6. Simplified 0.8

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

8. Applied add-cube-cbrt 0.8

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

10. Applied add-cube-cbrt 0.8

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

11. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020060 
(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))))