Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.0 → 1.9

Time: 35.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.794266843434528135733297660501734721002 \cdot 10^{-157}:\\ \;\;\;\;\frac{-x \cdot 2}{\left(-\frac{x}{y}\right) + 1}\\ \mathbf{elif}\;y \le 3.448485370032302944418093156077181588583 \cdot 10^{-155}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{x - y}\right)}^{3}} \cdot \left(2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array}\]

C

double f(double x, double y) {
        double r428514 = x;
        double r428515 = 2.0;
        double r428516 = r428514 * r428515;
        double r428517 = y;
        double r428518 = r428516 * r428517;
        double r428519 = r428514 - r428517;
        double r428520 = r428518 / r428519;
        return r428520;
}

↓

double f(double x, double y) {
        double r428521 = y;
        double r428522 = -1.794266843434528e-157;
        bool r428523 = r428521 <= r428522;
        double r428524 = x;
        double r428525 = 2.0;
        double r428526 = r428524 * r428525;
        double r428527 = -r428526;
        double r428528 = r428524 / r428521;
        double r428529 = -r428528;
        double r428530 = 1.0;
        double r428531 = r428529 + r428530;
        double r428532 = r428527 / r428531;
        double r428533 = 3.448485370032303e-155;
        bool r428534 = r428521 <= r428533;
        double r428535 = r428524 - r428521;
        double r428536 = r428524 / r428535;
        double r428537 = 3.0;
        double r428538 = pow(r428536, r428537);
        double r428539 = cbrt(r428538);
        double r428540 = r428525 * r428521;
        double r428541 = r428539 * r428540;
        double r428542 = r428535 / r428521;
        double r428543 = r428526 / r428542;
        double r428544 = r428534 ? r428541 : r428543;
        double r428545 = r428523 ? r428532 : r428544;
        return r428545;
}

Error

Bits error versus x

Bits error versus y

Target

Original	15.0
Target	0.3
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;x \lt -1.721044263414944729490876394165887012892 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x \lt 83645045635564432:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -1.794266843434528e-157

   1. Initial program 13.4

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

   2. Simplified 2.4

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

   4. Applied frac-2neg 2.4

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

   5. Simplified 2.4

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

   if -1.794266843434528e-157 < y < 3.448485370032303e-155

   1. Initial program 21.1

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

   2. Simplified 22.8

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

   4. Applied div-inv 22.9

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

   5. Applied times-frac 0.2

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

   6. Simplified 0.0

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

   8. Applied add-cbrt-cube 41.1

      \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}}} \cdot \left(2 \cdot y\right)\]

   9. Applied add-cbrt-cube 42.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \left(2 \cdot y\right)\]

   10. Applied cbrt-undiv 42.2

       \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(x \cdot x\right) \cdot x}{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}}} \cdot \left(2 \cdot y\right)\]

   11. Simplified 0.7

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x}{x - y}\right)}^{3}}} \cdot \left(2 \cdot y\right)\]

   if 3.448485370032303e-155 < y

   1. Initial program 12.6

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

   2. Simplified 2.3

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

   4. Applied *-un-lft-identity 2.3

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

   5. Applied *-un-lft-identity 2.3

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

   6. Applied times-frac 2.3

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

   7. Applied associate-/r* 2.3

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

   8. Simplified 2.3

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.794266843434528135733297660501734721002 \cdot 10^{-157}:\\ \;\;\;\;\frac{-x \cdot 2}{\left(-\frac{x}{y}\right) + 1}\\ \mathbf{elif}\;y \le 3.448485370032302944418093156077181588583 \cdot 10^{-155}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{x - y}\right)}^{3}} \cdot \left(2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 8.364504563556443e+16) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))