simple fma test

Average Error: 45.6 → 45.5

Time: 14.9s

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 r70704 = x;
        double r70705 = y;
        double r70706 = z;
        double r70707 = fma(r70704, r70705, r70706);
        double r70708 = 1.0;
        double r70709 = r70704 * r70705;
        double r70710 = r70709 + r70706;
        double r70711 = r70708 + r70710;
        double r70712 = r70707 - r70711;
        return r70712;
}

↓

double f(double x, double y, double z) {
        double r70713 = x;
        double r70714 = y;
        double r70715 = z;
        double r70716 = fma(r70713, r70714, r70715);
        double r70717 = 1.0;
        double r70718 = r70716 - r70717;
        double r70719 = r70713 * r70714;
        double r70720 = r70718 - r70719;
        double r70721 = r70720 - r70715;
        double r70722 = cbrt(r70721);
        double r70723 = r70722 * r70722;
        double r70724 = r70722 * r70723;
        return r70724;
}

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))))