simple fma test

Average Error: 45.3 → 45.3

Time: 5.0s

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 r64847 = x;
        double r64848 = y;
        double r64849 = z;
        double r64850 = fma(r64847, r64848, r64849);
        double r64851 = 1.0;
        double r64852 = r64847 * r64848;
        double r64853 = r64852 + r64849;
        double r64854 = r64851 + r64853;
        double r64855 = r64850 - r64854;
        return r64855;
}

↓

double f(double x, double y, double z) {
        double r64856 = x;
        double r64857 = y;
        double r64858 = z;
        double r64859 = fma(r64856, r64857, r64858);
        double r64860 = 1.0;
        double r64861 = r64856 * r64857;
        double r64862 = r64861 + r64858;
        double r64863 = r64860 + r64862;
        double r64864 = r64859 - r64863;
        double r64865 = cbrt(r64864);
        double r64866 = r64865 * r64865;
        double r64867 = r64866 * r64865;
        return r64867;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.3
Target	0
Herbie	45.3

\[-1\]

Derivation

1. Initial program 45.3

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

3. Applied add-cube-cbrt 45.3

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

4. Final simplification 45.3

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

Reproduce

herbie shell --seed 2020020 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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