math.cube on complex, imaginary part

Average Error: 7.3 → 0.2

Time: 27.0s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8262342 = x_re;
        double r8262343 = r8262342 * r8262342;
        double r8262344 = x_im;
        double r8262345 = r8262344 * r8262344;
        double r8262346 = r8262343 - r8262345;
        double r8262347 = r8262346 * r8262344;
        double r8262348 = r8262342 * r8262344;
        double r8262349 = r8262344 * r8262342;
        double r8262350 = r8262348 + r8262349;
        double r8262351 = r8262350 * r8262342;
        double r8262352 = r8262347 + r8262351;
        return r8262352;
}

↓

double f(double x_re, double x_im) {
        double r8262353 = x_re;
        double r8262354 = x_im;
        double r8262355 = 3.0;
        double r8262356 = r8262354 * r8262355;
        double r8262357 = r8262356 * r8262353;
        double r8262358 = r8262353 * r8262357;
        double r8262359 = pow(r8262354, r8262355);
        double r8262360 = r8262358 - r8262359;
        return r8262360;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.3
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 7.3

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

2. Taylor expanded around 0 7.2

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

3. Simplified 0.2

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

4. Taylor expanded around 0 7.3

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

5. Simplified 0.2

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

7. Applied pow1 0.2

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

8. Applied pow1 0.2

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

9. Applied pow-prod-up 0.2

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

10. Applied pow1 0.2

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

11. Applied pow-prod-up 0.2

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

12. Simplified 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(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)))