Linear.Projection:inversePerspective from linear-1.19.1.3, B

Average Error: 14.7 → 0.2

Time: 8.3s

Precision: 64

Math

\[\frac{x - y}{\left(x \cdot 2\right) \cdot y}\]

↓

\[\mathsf{fma}\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}, \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}, -\frac{1}{2} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\frac{1}{2}, \frac{1}{x}, \frac{1}{2} \cdot \frac{1}{x}\right)\]

C

double code(double x, double y) {
	return ((double) (((double) (x - y)) / ((double) (((double) (x * 2.0)) * y))));
}

↓

double code(double x, double y) {
	return ((double) (((double) fma(((double) (((double) (1.0 / ((double) sqrt(((double) sqrt(2.0)))))) / ((double) sqrt(((double) sqrt(2.0)))))), ((double) (((double) cbrt(1.0)) / ((double) (((double) sqrt(2.0)) * y)))), ((double) -(((double) (((double) (1.0 / 2.0)) * ((double) (1.0 / x)))))))) + ((double) fma(((double) -(((double) (1.0 / 2.0)))), ((double) (1.0 / x)), ((double) (((double) (1.0 / 2.0)) * ((double) (1.0 / x))))))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	14.7
Target	0.0
Herbie	0.2

\[\frac{0.5}{y} - \frac{0.5}{x}\]

Derivation

1. Initial program 14.7

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y}\]
2. Using strategy rm

3. Applied div-sub 14.7

   \[\leadsto \color{blue}{\frac{x}{\left(x \cdot 2\right) \cdot y} - \frac{y}{\left(x \cdot 2\right) \cdot y}}\]

4. Simplified 11.0

   \[\leadsto \color{blue}{\frac{1}{2 \cdot y}} - \frac{y}{\left(x \cdot 2\right) \cdot y}\]

5. Simplified 0.0

   \[\leadsto \frac{1}{2 \cdot y} - \color{blue}{\frac{1}{x \cdot 2}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.0

   \[\leadsto \frac{1}{2 \cdot y} - \frac{\color{blue}{1 \cdot 1}}{x \cdot 2}\]

8. Applied times-frac 0.0

   \[\leadsto \frac{1}{2 \cdot y} - \color{blue}{\frac{1}{x} \cdot \frac{1}{2}}\]

9. Applied add-sqr-sqrt 0.6

   \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot y} - \frac{1}{x} \cdot \frac{1}{2}\]

10. Applied associate-*l* 0.4

    \[\leadsto \frac{1}{\color{blue}{\sqrt{2} \cdot \left(\sqrt{2} \cdot y\right)}} - \frac{1}{x} \cdot \frac{1}{2}\]

11. Applied add-cube-cbrt 0.4

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{2} \cdot \left(\sqrt{2} \cdot y\right)} - \frac{1}{x} \cdot \frac{1}{2}\]

12. Applied times-frac 0.5

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{2}} \cdot \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}} - \frac{1}{x} \cdot \frac{1}{2}\]

13. Applied prod-diff 0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{2}}, \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}, -\frac{1}{2} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\frac{1}{2}, \frac{1}{x}, \frac{1}{2} \cdot \frac{1}{x}\right)}\]
14. Using strategy rm

15. Applied add-sqr-sqrt 0.5

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}, \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}, -\frac{1}{2} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\frac{1}{2}, \frac{1}{x}, \frac{1}{2} \cdot \frac{1}{x}\right)\]

16. Applied sqrt-prod 0.2

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}, \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}, -\frac{1}{2} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\frac{1}{2}, \frac{1}{x}, \frac{1}{2} \cdot \frac{1}{x}\right)\]

17. Applied associate-/r* 0.2

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}, \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}, -\frac{1}{2} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\frac{1}{2}, \frac{1}{x}, \frac{1}{2} \cdot \frac{1}{x}\right)\]

18. Simplified 0.2

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{2}}}, \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}, -\frac{1}{2} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\frac{1}{2}, \frac{1}{x}, \frac{1}{2} \cdot \frac{1}{x}\right)\]

19. Final simplification 0.2

    \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}, \frac{\sqrt[3]{1}}{\sqrt{2} \cdot y}, -\frac{1}{2} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\frac{1}{2}, \frac{1}{x}, \frac{1}{2} \cdot \frac{1}{x}\right)\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (- (/ 0.5 y) (/ 0.5 x))

  (/ (- x y) (* (* x 2) y)))