math.cube on complex, imaginary part

Average Error: 7.0 → 0.2

Time: 12.6s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\right)\]

C

double f(double x_re, double x_im) {
        double r1334454 = x_re;
        double r1334455 = r1334454 * r1334454;
        double r1334456 = x_im;
        double r1334457 = r1334456 * r1334456;
        double r1334458 = r1334455 - r1334457;
        double r1334459 = r1334458 * r1334456;
        double r1334460 = r1334454 * r1334456;
        double r1334461 = r1334456 * r1334454;
        double r1334462 = r1334460 + r1334461;
        double r1334463 = r1334462 * r1334454;
        double r1334464 = r1334459 + r1334463;
        return r1334464;
}

↓

double f(double x_re, double x_im) {
        double r1334465 = x_im;
        double r1334466 = x_re;
        double r1334467 = r1334465 + r1334466;
        double r1334468 = r1334466 - r1334465;
        double r1334469 = r1334468 * r1334465;
        double r1334470 = r1334466 * r1334465;
        double r1334471 = r1334470 + r1334470;
        double r1334472 = r1334471 * r1334466;
        double r1334473 = fma(r1334467, r1334469, r1334472);
        return r1334473;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 7.0

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

3. Applied difference-of-squares 7.0

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

4. Applied associate-*l* 0.2

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

6. Applied fma-def 0.2

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

7. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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