simple fma test

Average Error: 45.0 → 44.9

Time: 14.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r37366 = x;
        double r37367 = y;
        double r37368 = z;
        double r37369 = fma(r37366, r37367, r37368);
        double r37370 = 1.0;
        double r37371 = r37366 * r37367;
        double r37372 = r37371 + r37368;
        double r37373 = r37370 + r37372;
        double r37374 = r37369 - r37373;
        return r37374;
}

↓

double f(double x, double y, double z) {
        double r37375 = x;
        double r37376 = y;
        double r37377 = z;
        double r37378 = fma(r37375, r37376, r37377);
        double r37379 = 1.0;
        double r37380 = r37378 - r37379;
        double r37381 = r37375 * r37376;
        double r37382 = r37380 - r37381;
        double r37383 = r37382 - r37377;
        double r37384 = cbrt(r37383);
        double r37385 = r37384 * r37384;
        double r37386 = r37385 * r37384;
        double r37387 = cbrt(r37386);
        double r37388 = r37385 * r37387;
        return r37388;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.0
Target	0
Herbie	44.9

\[-1\]

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 associate--r+ 45.0

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

5. Applied associate--r+ 44.9

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

7. Applied add-cube-cbrt 44.9

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

9. Applied add-cube-cbrt 44.9

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

10. Final simplification 44.9

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

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1.0

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