Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.8 → 0.2

Time: 1.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.390909119607474 \cdot 10^{-45} \lor \neg \left(x \le 7.09985068380016666 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{x}{x - y} \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(y \cdot \frac{1}{x - y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r636326 = x;
        double r636327 = 2.0;
        double r636328 = r636326 * r636327;
        double r636329 = y;
        double r636330 = r636328 * r636329;
        double r636331 = r636326 - r636329;
        double r636332 = r636330 / r636331;
        return r636332;
}

↓

double f(double x, double y) {
        double r636333 = x;
        double r636334 = -1.390909119607474e-45;
        bool r636335 = r636333 <= r636334;
        double r636336 = 7.099850683800167e-27;
        bool r636337 = r636333 <= r636336;
        double r636338 = !r636337;
        bool r636339 = r636335 || r636338;
        double r636340 = y;
        double r636341 = r636333 - r636340;
        double r636342 = r636333 / r636341;
        double r636343 = 2.0;
        double r636344 = r636340 * r636343;
        double r636345 = r636342 * r636344;
        double r636346 = r636333 * r636343;
        double r636347 = 1.0;
        double r636348 = r636347 / r636341;
        double r636349 = r636340 * r636348;
        double r636350 = r636346 * r636349;
        double r636351 = r636339 ? r636345 : r636350;
        return r636351;
}

Error

Bits error versus x

Bits error versus y

Target

Original	14.8
Target	0.4
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x \lt -1.7210442634149447 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x \lt 83645045635564432:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.390909119607474e-45 or 7.099850683800167e-27 < x

   1. Initial program 14.4

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

   3. Applied associate-/l* 13.0

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

   5. Applied div-inv 13.2

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

   6. Applied times-frac 0.4

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

   7. Simplified 0.3

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

   if -1.390909119607474e-45 < x < 7.099850683800167e-27

   1. Initial program 15.4

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

   3. Applied *-un-lft-identity 15.4

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

   4. Applied times-frac 0.0

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

   5. Simplified 0.0

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

   7. Applied div-inv 0.2

      \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\left(y \cdot \frac{1}{x - y}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.390909119607474 \cdot 10^{-45} \lor \neg \left(x \le 7.09985068380016666 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{x}{x - y} \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(y \cdot \frac{1}{x - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564432) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y)))

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