simple fma test

Average Error: 45.1 → 45.0

Time: 13.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r47605 = x;
        double r47606 = y;
        double r47607 = z;
        double r47608 = fma(r47605, r47606, r47607);
        double r47609 = 1.0;
        double r47610 = r47605 * r47606;
        double r47611 = r47610 + r47607;
        double r47612 = r47609 + r47611;
        double r47613 = r47608 - r47612;
        return r47613;
}

↓

double f(double x, double y, double z) {
        double r47614 = x;
        double r47615 = y;
        double r47616 = z;
        double r47617 = fma(r47614, r47615, r47616);
        double r47618 = 1.0;
        double r47619 = r47617 - r47618;
        double r47620 = r47614 * r47615;
        double r47621 = r47619 - r47620;
        double r47622 = r47621 - r47616;
        double r47623 = cbrt(r47622);
        double r47624 = r47623 * r47623;
        double r47625 = r47624 * r47623;
        double r47626 = cbrt(r47625);
        double r47627 = r47624 * r47626;
        return r47627;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.1
Target	0
Herbie	45.0

\[-1\]

Derivation

1. Initial program 45.1

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

3. Applied associate--r+ 45.1

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

5. Applied associate--r+ 45.0

   \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}\]
6. Using strategy rm

7. Applied add-cube-cbrt 45.0

   \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}\right) \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}}\]
8. Using strategy rm

9. Applied add-cube-cbrt 45.0

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

10. Final simplification 45.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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