Average Error: 45.7 → 46.2
Time: 5.1s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \frac{z + x \cdot y}{\sqrt[3]{x \cdot y - z} \cdot \sqrt[3]{x \cdot y - z}} \cdot \frac{x \cdot y - z}{\sqrt[3]{x \cdot y - z}}\right)\]
\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{z + x \cdot y}{\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)
(FPCore (x y z) :precision binary64 (- (fma x y z) (+ 1.0 (+ (* x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (-
  (fma x y z)
  (+
   1.0
   (*
    (/ (+ z (* x y)) (* (cbrt (- (* x y) z)) (cbrt (- (* x y) z))))
    (/ (- (* x y) z) (cbrt (- (* x y) z)))))))
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) (z + ((double) (x * y)))) / ((double) (((double) cbrt(((double) (((double) (x * y)) - z)))) * ((double) cbrt(((double) (((double) (x * y)) - z))))))) * (((double) (((double) (x * y)) - z)) / ((double) cbrt(((double) (((double) (x * y)) - z)))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.7
Target0
Herbie46.2
\[-1\]

Derivation

  1. Initial program 45.7

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

  3. Applied flip-+_binary6446.4

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

  5. Applied add-cube-cbrt_binary6446.8

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

  6. Applied difference-of-squares_binary6446.8

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

  7. Applied times-frac_binary6446.2

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

  8. Final simplification46.2

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

Reproduce

herbie shell --seed 2020210 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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