math.cube on complex, imaginary part

Average Error: 7.1 → 0.6

Time: 24.1s

Precision: 64

Math

\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

↓

\[\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \sqrt[3]{x.re} \cdot \left(\left(\sqrt[3]{x.re} \cdot \sqrt[3]{x.re}\right) \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)\]

C

double f(double x_re, double x_im) {
        double r6593508 = x_re;
        double r6593509 = r6593508 * r6593508;
        double r6593510 = x_im;
        double r6593511 = r6593510 * r6593510;
        double r6593512 = r6593509 - r6593511;
        double r6593513 = r6593512 * r6593510;
        double r6593514 = r6593508 * r6593510;
        double r6593515 = r6593510 * r6593508;
        double r6593516 = r6593514 + r6593515;
        double r6593517 = r6593516 * r6593508;
        double r6593518 = r6593513 + r6593517;
        return r6593518;
}

↓

double f(double x_re, double x_im) {
        double r6593519 = x_re;
        double r6593520 = x_im;
        double r6593521 = r6593519 - r6593520;
        double r6593522 = r6593521 * r6593520;
        double r6593523 = r6593520 + r6593519;
        double r6593524 = r6593522 * r6593523;
        double r6593525 = cbrt(r6593519);
        double r6593526 = r6593525 * r6593525;
        double r6593527 = r6593519 * r6593520;
        double r6593528 = r6593527 + r6593527;
        double r6593529 = r6593526 * r6593528;
        double r6593530 = r6593525 * r6593529;
        double r6593531 = r6593524 + r6593530;
        return r6593531;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.3
Herbie	0.6

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

1. Initial program 7.1

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
2. Using strategy rm

3. Applied difference-of-squares 7.1

   \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

4. Applied associate-*l* 0.3

   \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
5. Using strategy rm

6. Applied add-cube-cbrt 0.6

   \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \color{blue}{\left(\left(\sqrt[3]{x.re} \cdot \sqrt[3]{x.re}\right) \cdot \sqrt[3]{x.re}\right)}\]

7. Applied associate-*r* 0.6

   \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(\sqrt[3]{x.re} \cdot \sqrt[3]{x.re}\right)\right) \cdot \sqrt[3]{x.re}}\]

8. Simplified 0.6

   \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot \left(\sqrt[3]{x.re} \cdot \sqrt[3]{x.re}\right)\right)} \cdot \sqrt[3]{x.re}\]

9. Final simplification 0.6

   \[\leadsto \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \sqrt[3]{x.re} \cdot \left(\left(\sqrt[3]{x.re} \cdot \sqrt[3]{x.re}\right) \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))