Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.9 → 0.1

Time: 26.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -817211854313671552:\\ \;\;\;\;\frac{1}{\frac{\frac{\frac{x}{y} - 1}{x}}{2}}\\ \mathbf{elif}\;y \le 4.039124438402000313213123565631371136853 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot 2}{x - y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \end{array}\]

C

double f(double x, double y) {
        double r174021 = x;
        double r174022 = 2.0;
        double r174023 = r174021 * r174022;
        double r174024 = y;
        double r174025 = r174023 * r174024;
        double r174026 = r174021 - r174024;
        double r174027 = r174025 / r174026;
        return r174027;
}

↓

double f(double x, double y) {
        double r174028 = y;
        double r174029 = -8.172118543136716e+17;
        bool r174030 = r174028 <= r174029;
        double r174031 = 1.0;
        double r174032 = x;
        double r174033 = r174032 / r174028;
        double r174034 = r174033 - r174031;
        double r174035 = r174034 / r174032;
        double r174036 = 2.0;
        double r174037 = r174035 / r174036;
        double r174038 = r174031 / r174037;
        double r174039 = 4.039124438402e-09;
        bool r174040 = r174028 <= r174039;
        double r174041 = r174032 * r174036;
        double r174042 = r174032 - r174028;
        double r174043 = r174041 / r174042;
        double r174044 = r174043 * r174028;
        double r174045 = r174041 / r174034;
        double r174046 = r174040 ? r174044 : r174045;
        double r174047 = r174030 ? r174038 : r174046;
        return r174047;
}

Error

Bits error versus x

Bits error versus y

Target

Original	14.9
Target	0.4
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -8.172118543136716e+17

   1. Initial program 17.3

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

   2. Simplified 0.1

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

   4. Applied *-un-lft-identity 0.1

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

   5. Applied *-un-lft-identity 0.1

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

   6. Applied times-frac 0.1

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

   7. Simplified 0.1

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

   8. Simplified 0.1

      \[\leadsto \frac{x \cdot 2}{1 \cdot \color{blue}{\left(\frac{x}{y} - 1\right)}}\]
   9. Using strategy rm

   10. Applied clear-num 0.2

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

   11. Simplified 0.2

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

   if -8.172118543136716e+17 < y < 4.039124438402e-09

   1. Initial program 13.3

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

   2. Simplified 15.0

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

   4. Applied *-un-lft-identity 15.0

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

   5. Applied *-un-lft-identity 15.0

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

   6. Applied times-frac 15.0

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

   7. Applied times-frac 15.1

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

   8. Simplified 15.1

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

   9. Simplified 14.2

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

   11. Applied associate-*r* 0.2

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

   12. Simplified 0.1

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

   if 4.039124438402e-09 < y

   1. Initial program 15.7

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

   2. Simplified 0.0

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

   4. Applied *-un-lft-identity 0.0

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

   5. Applied *-un-lft-identity 0.0

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

   6. Applied times-frac 0.0

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

   7. Simplified 0.0

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

   8. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -817211854313671552:\\ \;\;\;\;\frac{1}{\frac{\frac{\frac{x}{y} - 1}{x}}{2}}\\ \mathbf{elif}\;y \le 4.039124438402000313213123565631371136853 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot 2}{x - y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))