simple fma test

Average Error: 45.6 → 45.5

Time: 15.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r54432 = x;
        double r54433 = y;
        double r54434 = z;
        double r54435 = fma(r54432, r54433, r54434);
        double r54436 = 1.0;
        double r54437 = r54432 * r54433;
        double r54438 = r54437 + r54434;
        double r54439 = r54436 + r54438;
        double r54440 = r54435 - r54439;
        return r54440;
}

↓

double f(double x, double y, double z) {
        double r54441 = x;
        double r54442 = y;
        double r54443 = z;
        double r54444 = fma(r54441, r54442, r54443);
        double r54445 = 1.0;
        double r54446 = r54444 - r54445;
        double r54447 = r54441 * r54442;
        double r54448 = r54446 - r54447;
        double r54449 = r54448 - r54443;
        double r54450 = cbrt(r54449);
        double r54451 = r54450 * r54450;
        double r54452 = r54450 * r54451;
        return r54452;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.6
Target	0
Herbie	45.5

\[-1\]

Derivation

1. Initial program 45.6

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

2. Simplified 45.6

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

4. Applied associate--r+ 45.5

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

6. Applied add-cube-cbrt 45.5

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

7. Final simplification 45.5

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1.0

  (- (fma x y z) (+ 1.0 (+ (* x y) z))))