Average Error: 45.1 → 45.1
Time: 5.6s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right) \]
↓
\[\sqrt[3]{{\left(\mathsf{fma}\left(x, y, z\right) - \left(z + \left(1 + x \cdot y\right)\right)\right)}^{3}} \]
\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(z + \left(1 + 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) (+ z (+ 1.0 (* 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) - (z + (1.0 + (x * y)))), 3.0));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

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-cube_binary6446.3

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

  4. Simplified46.3

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

  6. Applied add-cbrt-cube_binary6446.3

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

  7. Simplified45.1

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

  8. Final simplification45.1

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

Reproduce

herbie shell --seed 2021206 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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