Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.2 → 0.2

Time: 1.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.7538803200316063 \cdot 10^{49} \lor \neg \left(x \le 2.59687826800368321 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{x - y} \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \end{array}\]

C

double f(double x, double y) {
        double r521518 = x;
        double r521519 = 2.0;
        double r521520 = r521518 * r521519;
        double r521521 = y;
        double r521522 = r521520 * r521521;
        double r521523 = r521518 - r521521;
        double r521524 = r521522 / r521523;
        return r521524;
}

↓

double f(double x, double y) {
        double r521525 = x;
        double r521526 = -1.7538803200316063e+49;
        bool r521527 = r521525 <= r521526;
        double r521528 = 2.5968782680036832e-23;
        bool r521529 = r521525 <= r521528;
        double r521530 = !r521529;
        bool r521531 = r521527 || r521530;
        double r521532 = y;
        double r521533 = r521525 - r521532;
        double r521534 = r521525 / r521533;
        double r521535 = 2.0;
        double r521536 = r521532 * r521535;
        double r521537 = r521534 * r521536;
        double r521538 = r521525 * r521535;
        double r521539 = r521525 / r521532;
        double r521540 = 1.0;
        double r521541 = r521539 - r521540;
        double r521542 = r521538 / r521541;
        double r521543 = r521531 ? r521537 : r521542;
        return r521543;
}

Error

Bits error versus x

Bits error versus y

Target

Original	15.2
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 < -1.7538803200316063e+49 or 2.5968782680036832e-23 < x

   1. Initial program 16.5

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

   3. Applied associate-/l* 15.6

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

   5. Applied div-inv 15.8

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

   6. Applied times-frac 0.3

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

   7. Simplified 0.1

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

   if -1.7538803200316063e+49 < x < 2.5968782680036832e-23

   1. Initial program 13.9

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

   3. Applied associate-/l* 0.2

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

   5. Applied div-sub 0.2

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

   6. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.7538803200316063 \cdot 10^{49} \lor \neg \left(x \le 2.59687826800368321 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{x - y} \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))