simple fma test

Average Error: 45.7 → 46.2

Time: 5.4s

Precision: 64

Math

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

↓

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

C

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) cbrt(((double) (((double) cbrt(((double) (((double) (x * y)) - z)))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (((double) (x * y)) - z)))) * ((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

Target

Original	45.7
Target	0
Herbie	46.2

\[-1\]

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-+ 46.4

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

5. Applied add-cube-cbrt 46.7

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

6. Applied difference-of-squares 46.7

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

7. Applied times-frac 46.2

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

9. Applied add-cube-cbrt 46.1

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

10. Applied cbrt-prod 46.2

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

12. Applied add-cube-cbrt 46.2

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

13. Applied cbrt-prod 46.2

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

14. Final simplification 46.2

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

Reproduce

herbie shell --seed 2020121 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))