simple fma test

Average Error: 45.4 → 45.3

Time: 19.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r55154 = x;
        double r55155 = y;
        double r55156 = z;
        double r55157 = fma(r55154, r55155, r55156);
        double r55158 = 1.0;
        double r55159 = r55154 * r55155;
        double r55160 = r55159 + r55156;
        double r55161 = r55158 + r55160;
        double r55162 = r55157 - r55161;
        return r55162;
}

↓

double f(double x, double y, double z) {
        double r55163 = x;
        double r55164 = y;
        double r55165 = z;
        double r55166 = fma(r55163, r55164, r55165);
        double r55167 = 1.0;
        double r55168 = r55166 - r55167;
        double r55169 = r55163 * r55164;
        double r55170 = r55168 - r55169;
        double r55171 = r55170 - r55165;
        double r55172 = cbrt(r55171);
        double r55173 = r55172 * r55172;
        double r55174 = r55173 * r55172;
        return r55174;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.4
Target	0
Herbie	45.3

\[-1\]

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 associate--r+ 45.4

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

5. Applied associate--r+ 45.3

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

7. Applied add-cube-cbrt 45.3

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

8. Final simplification 45.3

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

Reproduce

herbie shell --seed 2019199 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1.0

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