math.cube on complex, real part

Average Error: 7.2 → 0.2

Time: 17.6s

Precision: 64

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

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r221585 = x_re;
        double r221586 = r221585 * r221585;
        double r221587 = x_im;
        double r221588 = r221587 * r221587;
        double r221589 = r221586 - r221588;
        double r221590 = r221589 * r221585;
        double r221591 = r221585 * r221587;
        double r221592 = r221587 * r221585;
        double r221593 = r221591 + r221592;
        double r221594 = r221593 * r221587;
        double r221595 = r221590 - r221594;
        return r221595;
}

↓

double f(double x_re, double x_im) {
        double r221596 = x_re;
        double r221597 = 3.0;
        double r221598 = pow(r221596, r221597);
        double r221599 = x_im;
        double r221600 = r221599 * r221597;
        double r221601 = r221599 * r221596;
        double r221602 = r221600 * r221601;
        double r221603 = r221598 - r221602;
        return r221603;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.2
Target	0.3
Herbie	0.2

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

Derivation

1. Initial program 7.2

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

2. Simplified 0.2

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

4. Applied pow1 0.2

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

5. Applied pow1 0.2

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

6. Applied pow1 0.2

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

7. Applied pow-prod-down 0.2

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

8. Applied pow-prod-down 0.2

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

9. Simplified 0.2

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

11. Applied unpow-prod-down 0.2

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

12. Applied associate-*r* 0.2

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

13. Simplified 0.2

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

14. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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