math.cube on complex, real part

Average Error: 7.1 → 0.2

Time: 19.6s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r7540482 = x_re;
        double r7540483 = r7540482 * r7540482;
        double r7540484 = x_im;
        double r7540485 = r7540484 * r7540484;
        double r7540486 = r7540483 - r7540485;
        double r7540487 = r7540486 * r7540482;
        double r7540488 = r7540482 * r7540484;
        double r7540489 = r7540484 * r7540482;
        double r7540490 = r7540488 + r7540489;
        double r7540491 = r7540490 * r7540484;
        double r7540492 = r7540487 - r7540491;
        return r7540492;
}

↓

double f(double x_re, double x_im) {
        double r7540493 = x_im;
        double r7540494 = -r7540493;
        double r7540495 = x_re;
        double r7540496 = r7540495 * r7540493;
        double r7540497 = r7540496 + r7540496;
        double r7540498 = r7540493 * r7540497;
        double r7540499 = fma(r7540494, r7540497, r7540498);
        double r7540500 = r7540493 + r7540495;
        double r7540501 = r7540495 - r7540493;
        double r7540502 = r7540501 * r7540495;
        double r7540503 = r7540494 * r7540497;
        double r7540504 = fma(r7540500, r7540502, r7540503);
        double r7540505 = r7540499 + r7540504;
        return r7540505;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
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 7.1

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
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* 0.2

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

6. Applied prod-diff 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019134 +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)))