Average Error: 45.0 → 45.6
Time: 5.2s
Precision: 64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \frac{x \cdot y + z}{\sqrt[3]{x \cdot y - z} \cdot \sqrt[3]{x \cdot y - z}} \cdot \frac{x \cdot y - z}{\left(\sqrt[3]{\sqrt[3]{x \cdot y - z}} \cdot \sqrt[3]{\sqrt[3]{x \cdot y - z}}\right) \cdot \sqrt[3]{\sqrt[3]{x \cdot y - z}}}\right)\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\mathsf{fma}\left(x, y, z\right) - \left(1 + \frac{x \cdot y + z}{\sqrt[3]{x \cdot y - z} \cdot \sqrt[3]{x \cdot y - z}} \cdot \frac{x \cdot y - z}{\left(\sqrt[3]{\sqrt[3]{x \cdot y - z}} \cdot \sqrt[3]{\sqrt[3]{x \cdot y - z}}\right) \cdot \sqrt[3]{\sqrt[3]{x \cdot y - z}}}\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) fma(x, y, z)) - ((double) (1.0 + ((double) (((double) (((double) (((double) (x * y)) + z)) / ((double) (((double) cbrt(((double) (((double) (x * y)) - z)))) * ((double) cbrt(((double) (((double) (x * y)) - z)))))))) * ((double) (((double) (((double) (x * y)) - z)) / ((double) (((double) (((double) cbrt(((double) cbrt(((double) (((double) (x * y)) - z)))))) * ((double) cbrt(((double) cbrt(((double) (((double) (x * y)) - z)))))))) * ((double) cbrt(((double) cbrt(((double) (((double) (x * y)) - z))))))))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Your Program's Arguments

Results

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Target

Original45.0
Target0
Herbie45.6
\[-1\]

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 flip-+45.8

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

  5. Applied add-cube-cbrt46.1

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

  6. Applied difference-of-squares46.1

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

  7. Applied times-frac45.5

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

  9. Applied add-cube-cbrt45.6

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

  10. Final simplification45.6

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

Reproduce

herbie shell --seed 2020126 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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