math.cube on complex, real part

Average Error: 7.5 → 0.2

Time: 17.5s

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 r7390466 = x_re;
        double r7390467 = r7390466 * r7390466;
        double r7390468 = x_im;
        double r7390469 = r7390468 * r7390468;
        double r7390470 = r7390467 - r7390469;
        double r7390471 = r7390470 * r7390466;
        double r7390472 = r7390466 * r7390468;
        double r7390473 = r7390468 * r7390466;
        double r7390474 = r7390472 + r7390473;
        double r7390475 = r7390474 * r7390468;
        double r7390476 = r7390471 - r7390475;
        return r7390476;
}

↓

double f(double x_re, double x_im) {
        double r7390477 = x_re;
        double r7390478 = x_im;
        double r7390479 = r7390477 - r7390478;
        double r7390480 = r7390479 * r7390477;
        double r7390481 = r7390478 + r7390477;
        double r7390482 = r7390480 * r7390481;
        double r7390483 = r7390477 * r7390478;
        double r7390484 = r7390483 + r7390483;
        double r7390485 = r7390484 * r7390478;
        double r7390486 = r7390482 - r7390485;
        return r7390486;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.5
Target	0.2
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 7.5

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

3. Applied difference-of-squares 7.5

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

4. Applied associate-*l* 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019192 +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 (* 3.0 x.im))))

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