math.cube on complex, real part

Average Error: 7.1 → 0.3

Time: 26.4s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r7108494 = x_re;
        double r7108495 = r7108494 * r7108494;
        double r7108496 = x_im;
        double r7108497 = r7108496 * r7108496;
        double r7108498 = r7108495 - r7108497;
        double r7108499 = r7108498 * r7108494;
        double r7108500 = r7108494 * r7108496;
        double r7108501 = r7108496 * r7108494;
        double r7108502 = r7108500 + r7108501;
        double r7108503 = r7108502 * r7108496;
        double r7108504 = r7108499 - r7108503;
        return r7108504;
}

↓

double f(double x_re, double x_im) {
        double r7108505 = x_re;
        double r7108506 = x_im;
        double r7108507 = r7108505 - r7108506;
        double r7108508 = r7108507 * r7108505;
        double r7108509 = r7108506 + r7108505;
        double r7108510 = r7108508 * r7108509;
        double r7108511 = r7108505 * r7108506;
        double r7108512 = r7108511 + r7108511;
        double r7108513 = r7108512 * r7108506;
        double r7108514 = r7108510 - r7108513;
        return r7108514;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 7.1

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

3. Applied difference-of-squares 7.1

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

4. Applied associate-*l* 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :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)))