Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.2 → 1.5

Time: 7.0s

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 r579598 = x;
        double r579599 = 2.0;
        double r579600 = r579598 * r579599;
        double r579601 = y;
        double r579602 = r579600 * r579601;
        double r579603 = r579598 - r579601;
        double r579604 = r579602 / r579603;
        return r579604;
}

↓

double f(double x, double y) {
        double r579605 = x;
        double r579606 = 2.0;
        double r579607 = r579605 * r579606;
        double r579608 = y;
        double r579609 = r579607 * r579608;
        double r579610 = r579605 - r579608;
        double r579611 = r579609 / r579610;
        double r579612 = -2.983744795196586e-65;
        bool r579613 = r579611 <= r579612;
        double r579614 = -8.215112199764979e-271;
        bool r579615 = r579611 <= r579614;
        double r579616 = 0.0;
        bool r579617 = r579611 <= r579616;
        double r579618 = !r579617;
        double r579619 = 6.505978645209074e-82;
        bool r579620 = r579611 <= r579619;
        bool r579621 = r579618 && r579620;
        bool r579622 = r579615 || r579621;
        double r579623 = !r579622;
        bool r579624 = r579613 || r579623;
        double r579625 = r579608 / r579610;
        double r579626 = r579607 * r579625;
        double r579627 = r579624 ? r579626 : r579611;
        return r579627;
}

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 +o rules:numerics
(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)))