Average Error: 45.0 → 45.0
Time: 1.9s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\sqrt[3]{{\left(\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right)\right)}^{3}}\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\sqrt[3]{{\left(\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right)\right)}^{3}}
(FPCore (x y z) :precision binary64 (- (fma x y z) (+ 1.0 (+ (* x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (cbrt (pow (- (fma x y z) (+ 1.0 (+ z (* x y)))) 3.0)))
double code(double x, double y, double z) {
	return fma(x, y, z) - (1.0 + ((x * y) + z));
}
↓
double code(double x, double y, double z) {
	return cbrt(pow((fma(x, y, z) - (1.0 + (z + (x * y)))), 3.0));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.0
Target0
Herbie45.0
\[-1\]

Derivation

  1. Initial program 45.0

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

  3. Applied add-cbrt-cube_binary64_56245.0

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

  4. Simplified45.0

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

  5. Final simplification45.0

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

Reproduce

herbie shell --seed 2020232 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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