simple fma test

Average Error: 45.4 → 45.4

Time: 5.1s

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 r65596 = x;
        double r65597 = y;
        double r65598 = z;
        double r65599 = fma(r65596, r65597, r65598);
        double r65600 = 1.0;
        double r65601 = r65596 * r65597;
        double r65602 = r65601 + r65598;
        double r65603 = r65600 + r65602;
        double r65604 = r65599 - r65603;
        return r65604;
}

↓

double f(double x, double y, double z) {
        double r65605 = x;
        double r65606 = y;
        double r65607 = z;
        double r65608 = fma(r65605, r65606, r65607);
        double r65609 = 1.0;
        double r65610 = r65605 * r65606;
        double r65611 = r65610 + r65607;
        double r65612 = r65609 + r65611;
        double r65613 = r65608 - r65612;
        double r65614 = cbrt(r65613);
        double r65615 = r65614 * r65614;
        double r65616 = r65615 * r65614;
        return r65616;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.4
Target	0
Herbie	45.4

\[-1\]

Derivation

1. Initial program 45.4

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

3. Applied add-cube-cbrt 45.4

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

4. Final simplification 45.4

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

Reproduce

herbie shell --seed 2020021 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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