Average Error: 45.1 → 7.9
Time: 2.9s
Precision: 64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\sqrt[3]{{\left(\left(\mathsf{fma}\left(x, y, z\right) + \left(-\left(z + x \cdot y\right)\right)\right) - 1\right)}^{3}}\]
\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) + \left(-\left(z + x \cdot y\right)\right)\right) - 1\right)}^{3}}
double f(double x, double y, double z) {
        double r75845 = x;
        double r75846 = y;
        double r75847 = z;
        double r75848 = fma(r75845, r75846, r75847);
        double r75849 = 1.0;
        double r75850 = r75845 * r75846;
        double r75851 = r75850 + r75847;
        double r75852 = r75849 + r75851;
        double r75853 = r75848 - r75852;
        return r75853;
}
↓
double f(double x, double y, double z) {
        double r75854 = x;
        double r75855 = y;
        double r75856 = z;
        double r75857 = fma(r75854, r75855, r75856);
        double r75858 = r75854 * r75855;
        double r75859 = r75856 + r75858;
        double r75860 = -r75859;
        double r75861 = r75857 + r75860;
        double r75862 = 1.0;
        double r75863 = r75861 - r75862;
        double r75864 = 3.0;
        double r75865 = pow(r75863, r75864);
        double r75866 = cbrt(r75865);
        return r75866;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.1
Target0
Herbie7.9
\[-1\]

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-cbrt-cube45.1

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

  4. Simplified45.1

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

  6. Applied associate--r+30.8

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

  8. Applied associate--r+14.9

    \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - x \cdot y\right) - 1\right)}}^{3}}\]
  9. Using strategy rm

  10. Applied sub-neg14.9

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

  11. Applied associate--l+7.9

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

  12. Simplified7.9

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

  13. Final simplification7.9

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

Reproduce

herbie shell --seed 2020027 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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