Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.2 → 1.5

Time: 6.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -2.98374479519658591 \cdot 10^{-65} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -8.21511219976497913 \cdot 10^{-271} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0\right) \land \frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 6.50597864520907364 \cdot 10^{-82}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \end{array}\]

C

double f(double x, double y) {
        double r606278 = x;
        double r606279 = 2.0;
        double r606280 = r606278 * r606279;
        double r606281 = y;
        double r606282 = r606280 * r606281;
        double r606283 = r606278 - r606281;
        double r606284 = r606282 / r606283;
        return r606284;
}

↓

double f(double x, double y) {
        double r606285 = x;
        double r606286 = 2.0;
        double r606287 = r606285 * r606286;
        double r606288 = y;
        double r606289 = r606287 * r606288;
        double r606290 = r606285 - r606288;
        double r606291 = r606289 / r606290;
        double r606292 = -2.983744795196586e-65;
        bool r606293 = r606291 <= r606292;
        double r606294 = -8.215112199764979e-271;
        bool r606295 = r606291 <= r606294;
        double r606296 = 0.0;
        bool r606297 = r606291 <= r606296;
        double r606298 = !r606297;
        double r606299 = 6.505978645209074e-82;
        bool r606300 = r606291 <= r606299;
        bool r606301 = r606298 && r606300;
        bool r606302 = r606295 || r606301;
        double r606303 = !r606302;
        bool r606304 = r606293 || r606303;
        double r606305 = r606288 / r606290;
        double r606306 = r606287 * r606305;
        double r606307 = r606304 ? r606306 : r606291;
        return r606307;
}

Error

Bits error versus x

Bits error versus y

Target

Original	15.2
Target	0.3
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (/ (* (* x 2.0) y) (- x y)) < -2.983744795196586e-65 or -8.215112199764979e-271 < (/ (* (* x 2.0) y) (- x y)) < 0.0 or 6.505978645209074e-82 < (/ (* (* x 2.0) y) (- x y))

   1. Initial program 28.0

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

   3. Applied *-un-lft-identity 28.0

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

   4. Applied times-frac 2.2

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

   5. Simplified 2.2

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

   if -2.983744795196586e-65 < (/ (* (* x 2.0) y) (- x y)) < -8.215112199764979e-271 or 0.0 < (/ (* (* x 2.0) y) (- x y)) < 6.505978645209074e-82

   1. Initial program 0.6

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -2.98374479519658591 \cdot 10^{-65} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -8.21511219976497913 \cdot 10^{-271} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0\right) \land \frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 6.50597864520907364 \cdot 10^{-82}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))