Linear.Projection:inversePerspective from linear-1.19.1.3, C

Average Error: 15.3 → 0.8

Time: 1.9s

Precision: binary64

Math

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

↓

\[\sqrt{0.5} \cdot \left(\frac{\sqrt{0.5}}{y} + \frac{\sqrt{0.5}}{x}\right)\]

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) sqrt(0.5)) * ((double) (((double) (((double) sqrt(0.5)) / y)) + ((double) (((double) sqrt(0.5)) / x))))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	15.3
Target	0.0
Herbie	0.8

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

Derivation

1. Initial program 15.3

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

2. Taylor expanded around 0 0.0

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x}}\]

3. Simplified 0.0

   \[\leadsto \color{blue}{\frac{0.5}{y} + \frac{0.5}{x}}\]
4. Using strategy rm

5. Applied *-un-lft-identity 0.0

   \[\leadsto \frac{0.5}{y} + \frac{0.5}{\color{blue}{1 \cdot x}}\]

6. Applied add-sqr-sqrt 0.6

   \[\leadsto \frac{0.5}{y} + \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{1 \cdot x}\]

7. Applied times-frac 0.4

   \[\leadsto \frac{0.5}{y} + \color{blue}{\frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{x}}\]

8. Applied *-un-lft-identity 0.4

   \[\leadsto \frac{0.5}{\color{blue}{1 \cdot y}} + \frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{x}\]

9. Applied add-sqr-sqrt 1.0

   \[\leadsto \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{1 \cdot y} + \frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{x}\]

10. Applied times-frac 0.8

    \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{y}} + \frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{x}\]

11. Applied distribute-lft-out 0.8

    \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{1} \cdot \left(\frac{\sqrt{0.5}}{y} + \frac{\sqrt{0.5}}{x}\right)}\]

12. Final simplification 0.8

    \[\leadsto \sqrt{0.5} \cdot \left(\frac{\sqrt{0.5}}{y} + \frac{\sqrt{0.5}}{x}\right)\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

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

  (/ (+ x y) (* (* x 2.0) y)))