Average Error: 45.1 → 8.1
Time: 3.0s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\sqrt[3]{{\left(\left(\mathsf{fma}\left(x, y, z\right) - \left(z + x \cdot y\right)\right) - 1\right)}^{3}}\]
\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(z + x \cdot y\right)\right) - 1\right)}^{3}}
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) cbrt(((double) pow(((double) (((double) (((double) fma(x, y, z)) - ((double) (z + ((double) (x * y)))))) - 1.0)), 3.0))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original45.1
Target0
Herbie8.1
\[-1\]

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+15.2

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

  10. Applied associate--l-8.1

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

  11. Final simplification8.1

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

Reproduce

herbie shell --seed 2020150 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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