Linear.Projection:inversePerspective from linear-1.19.1.3, C

Average Error: 15.5 → 0.7

Time: 4.6min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.3802292164302602 \cdot 10^{113} \lor \neg \left(y \le 7.44944977064311927 \cdot 10^{36}\right):\\ \;\;\;\;\frac{1}{x \cdot 2} \cdot \frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{y}{x} + 0.5}{y}\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if (((y <= -2.38022921643026e+113) || !(y <= 7.449449770643119e+36))) {
		VAR = ((double) ((1.0 / ((double) (x * 2.0))) * (((double) (x + y)) / y)));
	} else {
		VAR = (((double) (((double) (0.5 * (y / x))) + 0.5)) / y);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	15.5
Target	0.0
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if y < -2.3802292164302602e113 or 7.44944977064311927e36 < y

   1. Initial program 18.9

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

   3. Applied *-un-lft-identity 18.9

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

   4. Applied times-frac 0.1

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

   if -2.3802292164302602e113 < y < 7.44944977064311927e36

   1. Initial program 13.5

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

   3. Applied associate-/r* 1.1

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

   4. Taylor expanded around 0 1.0

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{y}{x} + 0.5}}{y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.3802292164302602 \cdot 10^{113} \lor \neg \left(y \le 7.44944977064311927 \cdot 10^{36}\right):\\ \;\;\;\;\frac{1}{x \cdot 2} \cdot \frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{y}{x} + 0.5}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))