Average Error: 44.9 → 44.7
Time: 13.5s
Precision: 64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}\right) \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}\]
\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(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}
double f(double x, double y, double z) {
        double r40751 = x;
        double r40752 = y;
        double r40753 = z;
        double r40754 = fma(r40751, r40752, r40753);
        double r40755 = 1.0;
        double r40756 = r40751 * r40752;
        double r40757 = r40756 + r40753;
        double r40758 = r40755 + r40757;
        double r40759 = r40754 - r40758;
        return r40759;
}
↓
double f(double x, double y, double z) {
        double r40760 = x;
        double r40761 = y;
        double r40762 = z;
        double r40763 = fma(r40760, r40761, r40762);
        double r40764 = 1.0;
        double r40765 = r40763 - r40764;
        double r40766 = r40760 * r40761;
        double r40767 = r40765 - r40766;
        double r40768 = r40767 - r40762;
        double r40769 = cbrt(r40768);
        double r40770 = r40769 * r40769;
        double r40771 = r40770 * r40769;
        return r40771;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original44.9
Target0
Herbie44.7
\[-1\]

Derivation

  1. Initial program 44.9

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

  3. Applied associate--r+44.9

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

  5. Applied associate--r+44.8

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

  7. Applied add-cube-cbrt44.7

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

  8. Final simplification44.7

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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