math.cube on complex, real part

Average Error: 7.4 → 0.3

Time: 20.3s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r13926482 = x_re;
        double r13926483 = r13926482 * r13926482;
        double r13926484 = x_im;
        double r13926485 = r13926484 * r13926484;
        double r13926486 = r13926483 - r13926485;
        double r13926487 = r13926486 * r13926482;
        double r13926488 = r13926482 * r13926484;
        double r13926489 = r13926484 * r13926482;
        double r13926490 = r13926488 + r13926489;
        double r13926491 = r13926490 * r13926484;
        double r13926492 = r13926487 - r13926491;
        return r13926492;
}

↓

double f(double x_re, double x_im) {
        double r13926493 = x_re;
        double r13926494 = x_im;
        double r13926495 = r13926493 + r13926494;
        double r13926496 = r13926493 - r13926494;
        double r13926497 = r13926496 * r13926493;
        double r13926498 = r13926495 * r13926497;
        double r13926499 = r13926494 * r13926493;
        double r13926500 = r13926499 + r13926499;
        double r13926501 = r13926500 * r13926494;
        double r13926502 = r13926498 - r13926501;
        return r13926502;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
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 7.4

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

3. Applied difference-of-squares 7.4

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

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

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

Reproduce

herbie shell --seed 2019174 
(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 (* 3.0 x.im))))

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