simple fma test

Average Error: 45.2 → 45.1

Time: 2.9s

Precision: 64

Math

\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]

↓

\[\left(\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\]

C

double f(double x, double y, double z) {
        double r75460 = x;
        double r75461 = y;
        double r75462 = z;
        double r75463 = fma(r75460, r75461, r75462);
        double r75464 = 1.0;
        double r75465 = r75460 * r75461;
        double r75466 = r75465 + r75462;
        double r75467 = r75464 + r75466;
        double r75468 = r75463 - r75467;
        return r75468;
}

↓

double f(double x, double y, double z) {
        double r75469 = x;
        double r75470 = y;
        double r75471 = z;
        double r75472 = fma(r75469, r75470, r75471);
        double r75473 = 1.0;
        double r75474 = r75469 * r75470;
        double r75475 = r75474 + r75471;
        double r75476 = r75473 + r75475;
        double r75477 = r75472 - r75476;
        double r75478 = cbrt(r75477);
        double r75479 = r75478 * r75478;
        double r75480 = r75479 * r75478;
        return r75480;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.2
Target	0
Herbie	45.1

\[-1\]

Derivation

1. Initial program 45.2

   \[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 45.1

   \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}}\]

4. Final simplification 45.1

   \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))