math.cube on complex, real part

Average Error: 7.1 → 0.3

Time: 22.2s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r6803208 = x_re;
        double r6803209 = r6803208 * r6803208;
        double r6803210 = x_im;
        double r6803211 = r6803210 * r6803210;
        double r6803212 = r6803209 - r6803211;
        double r6803213 = r6803212 * r6803208;
        double r6803214 = r6803208 * r6803210;
        double r6803215 = r6803210 * r6803208;
        double r6803216 = r6803214 + r6803215;
        double r6803217 = r6803216 * r6803210;
        double r6803218 = r6803213 - r6803217;
        return r6803218;
}

↓

double f(double x_re, double x_im) {
        double r6803219 = x_re;
        double r6803220 = x_im;
        double r6803221 = r6803219 - r6803220;
        double r6803222 = r6803221 * r6803219;
        double r6803223 = r6803220 + r6803219;
        double r6803224 = r6803222 * r6803223;
        double r6803225 = r6803219 * r6803220;
        double r6803226 = r6803225 + r6803225;
        double r6803227 = r6803226 * r6803220;
        double r6803228 = r6803224 - r6803227;
        return r6803228;
}

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 2019172 
(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)))