simple fma test

Average Error: 45.1 → 45.1

Time: 15.4s

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 r48744 = x;
        double r48745 = y;
        double r48746 = z;
        double r48747 = fma(r48744, r48745, r48746);
        double r48748 = 1.0;
        double r48749 = r48744 * r48745;
        double r48750 = r48749 + r48746;
        double r48751 = r48748 + r48750;
        double r48752 = r48747 - r48751;
        return r48752;
}

↓

double f(double x, double y, double z) {
        double r48753 = x;
        double r48754 = y;
        double r48755 = z;
        double r48756 = fma(r48753, r48754, r48755);
        double r48757 = 1.0;
        double r48758 = r48753 * r48754;
        double r48759 = r48758 + r48755;
        double r48760 = r48757 + r48759;
        double r48761 = r48756 - r48760;
        double r48762 = cbrt(r48761);
        double r48763 = r48762 * r48762;
        double r48764 = r48763 * r48762;
        return r48764;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.1
Target	0
Herbie	45.1

\[-1\]

Derivation

1. Initial program 45.1

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

3. Applied add-cube-cbrt 45.1

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

4. Final simplification 45.1

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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