Average Error: 44.8 → 30.8
Time: 4.2s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - z} \cdot \left(\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - z} \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - z}\right) - \left(1 + x \cdot y\right)\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - z} \cdot \left(\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - z} \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - z}\right) - \left(1 + x \cdot y\right)
double code(double x, double y, double z) {
	return ((double) (((double) fma(x, y, z)) - ((double) (1.0 + ((double) (((double) (x * y)) + z))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (((double) cbrt(((double) (((double) fma(x, y, z)) - z)))) * ((double) (((double) cbrt(((double) (((double) fma(x, y, z)) - z)))) * ((double) cbrt(((double) (((double) fma(x, y, z)) - z)))))))) - ((double) (1.0 + ((double) (x * y))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original44.8
Target0
Herbie30.8
\[-1\]

Derivation

  1. Initial program 44.8

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

  3. Applied add-cbrt-cube44.8

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

  4. Simplified44.8

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

  6. Applied associate--r+30.6

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

  8. Applied rem-cbrt-cube30.5

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

  10. Applied add-cube-cbrt30.8

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

  11. Final simplification30.8

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

Reproduce

herbie shell --seed 2020196 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

  (- (fma x y z) (+ 1.0 (+ (* x y) z))))