math.cube on complex, real part

Average Error: 7.7 → 0.2

Time: 27.2s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r164593 = x_re;
        double r164594 = r164593 * r164593;
        double r164595 = x_im;
        double r164596 = r164595 * r164595;
        double r164597 = r164594 - r164596;
        double r164598 = r164597 * r164593;
        double r164599 = r164593 * r164595;
        double r164600 = r164595 * r164593;
        double r164601 = r164599 + r164600;
        double r164602 = r164601 * r164595;
        double r164603 = r164598 - r164602;
        return r164603;
}

↓

double f(double x_re, double x_im) {
        double r164604 = x_re;
        double r164605 = 3.0;
        double r164606 = pow(r164604, r164605);
        double r164607 = x_im;
        double r164608 = r164607 * r164605;
        double r164609 = r164608 * r164604;
        double r164610 = r164609 * r164607;
        double r164611 = r164606 - r164610;
        return r164611;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.7
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 7.7

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

2. Simplified 0.2

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

4. Applied associate-*r* 0.2

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

6. Applied associate-*r* 0.2

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

7. Simplified 0.2

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

8. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019325 
(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)))