simple fma test

Average Error: 45.3 → 45.3

Time: 4.5s

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 r66745 = x;
        double r66746 = y;
        double r66747 = z;
        double r66748 = fma(r66745, r66746, r66747);
        double r66749 = 1.0;
        double r66750 = r66745 * r66746;
        double r66751 = r66750 + r66747;
        double r66752 = r66749 + r66751;
        double r66753 = r66748 - r66752;
        return r66753;
}

↓

double f(double x, double y, double z) {
        double r66754 = x;
        double r66755 = y;
        double r66756 = z;
        double r66757 = fma(r66754, r66755, r66756);
        double r66758 = 1.0;
        double r66759 = r66754 * r66755;
        double r66760 = r66759 + r66756;
        double r66761 = r66758 + r66760;
        double r66762 = r66757 - r66761;
        double r66763 = cbrt(r66762);
        double r66764 = r66763 * r66763;
        double r66765 = r66764 * r66763;
        return r66765;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.3
Target	0
Herbie	45.3

\[-1\]

Derivation

1. Initial program 45.3

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

3. Applied add-cube-cbrt 45.3

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

4. Final simplification 45.3

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

Reproduce

herbie shell --seed 2020024 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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