simple fma test

Average Error: 45.0 → 45.0

Time: 5.2s

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 r55853 = x;
        double r55854 = y;
        double r55855 = z;
        double r55856 = fma(r55853, r55854, r55855);
        double r55857 = 1.0;
        double r55858 = r55853 * r55854;
        double r55859 = r55858 + r55855;
        double r55860 = r55857 + r55859;
        double r55861 = r55856 - r55860;
        return r55861;
}

↓

double f(double x, double y, double z) {
        double r55862 = x;
        double r55863 = y;
        double r55864 = z;
        double r55865 = fma(r55862, r55863, r55864);
        double r55866 = 1.0;
        double r55867 = r55862 * r55863;
        double r55868 = r55867 + r55864;
        double r55869 = r55866 + r55868;
        double r55870 = r55865 - r55869;
        double r55871 = cbrt(r55870);
        double r55872 = r55871 * r55871;
        double r55873 = r55872 * r55871;
        return r55873;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.0
Target	0
Herbie	45.0

\[-1\]

Derivation

1. Initial program 45.0

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

3. Applied add-cube-cbrt 45.0

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

4. Final simplification 45.0

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

Reproduce

herbie shell --seed 2020003 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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