simple fma test

Average Error: 44.4 → 44.4

Time: 5.8s

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 r71749 = x;
        double r71750 = y;
        double r71751 = z;
        double r71752 = fma(r71749, r71750, r71751);
        double r71753 = 1.0;
        double r71754 = r71749 * r71750;
        double r71755 = r71754 + r71751;
        double r71756 = r71753 + r71755;
        double r71757 = r71752 - r71756;
        return r71757;
}

↓

double f(double x, double y, double z) {
        double r71758 = x;
        double r71759 = y;
        double r71760 = z;
        double r71761 = fma(r71758, r71759, r71760);
        double r71762 = 1.0;
        double r71763 = r71758 * r71759;
        double r71764 = r71763 + r71760;
        double r71765 = r71762 + r71764;
        double r71766 = r71761 - r71765;
        double r71767 = cbrt(r71766);
        double r71768 = r71767 * r71767;
        double r71769 = r71768 * r71767;
        return r71769;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	44.4
Target	0
Herbie	44.4

\[-1\]

Derivation

1. Initial program 44.4

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

3. Applied add-cube-cbrt 44.4

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

4. Final simplification 44.4

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

Reproduce

herbie shell --seed 2020035 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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