math.cube on complex, imaginary part

Average Error: 7.7 → 0.2

Time: 16.0s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8226414 = x_re;
        double r8226415 = r8226414 * r8226414;
        double r8226416 = x_im;
        double r8226417 = r8226416 * r8226416;
        double r8226418 = r8226415 - r8226417;
        double r8226419 = r8226418 * r8226416;
        double r8226420 = r8226414 * r8226416;
        double r8226421 = r8226416 * r8226414;
        double r8226422 = r8226420 + r8226421;
        double r8226423 = r8226422 * r8226414;
        double r8226424 = r8226419 + r8226423;
        return r8226424;
}

↓

double f(double x_re, double x_im) {
        double r8226425 = x_re;
        double r8226426 = x_im;
        double r8226427 = r8226425 * r8226426;
        double r8226428 = 3.0;
        double r8226429 = r8226427 * r8226428;
        double r8226430 = r8226429 * r8226425;
        double r8226431 = pow(r8226426, r8226428);
        double r8226432 = r8226430 - r8226431;
        return r8226432;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.7
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 7.7

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

3. Applied difference-of-squares 7.7

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

4. Applied associate-*l* 0.2

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

5. Taylor expanded around 0 0.2

   \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\left(2 \cdot \left(x.im \cdot x.re\right)\right)} \cdot x.re\]

6. Taylor expanded around 0 7.6

   \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot {x.re}^{2}\right) - {x.im}^{3}}\]

7. Simplified 0.2

   \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot x.im}\]
8. Using strategy rm

9. Applied pow1 0.2

   \[\leadsto x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \color{blue}{{x.im}^{1}}\]

10. Applied pow1 0.2

    \[\leadsto x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot \color{blue}{{x.im}^{1}}\right) \cdot {x.im}^{1}\]

11. Applied pow1 0.2

    \[\leadsto x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - \left(\color{blue}{{x.im}^{1}} \cdot {x.im}^{1}\right) \cdot {x.im}^{1}\]

12. Applied pow-prod-up 0.2

    \[\leadsto x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - \color{blue}{{x.im}^{\left(1 + 1\right)}} \cdot {x.im}^{1}\]

13. Applied pow-prod-up 0.2

    \[\leadsto x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - \color{blue}{{x.im}^{\left(\left(1 + 1\right) + 1\right)}}\]

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))