Average Error: 15.1 → 0.5
Time: 7.0s
Precision: 64
\[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.819157750295796344692295718235260807105 \cdot 10^{-37} \lor \neg \left(y \le 3.126349286155401959304305196339847373255 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\frac{1 + \frac{x}{y}}{2}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{y} \cdot \frac{x + y}{x}\\ \end{array}\]
\frac{x + y}{\left(x \cdot 2\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.819157750295796344692295718235260807105 \cdot 10^{-37} \lor \neg \left(y \le 3.126349286155401959304305196339847373255 \cdot 10^{-87}\right):\\
\;\;\;\;\frac{\frac{1 + \frac{x}{y}}{2}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2}}{y} \cdot \frac{x + y}{x}\\

\end{array}
double f(double x, double y) {
        double r337635 = x;
        double r337636 = y;
        double r337637 = r337635 + r337636;
        double r337638 = 2.0;
        double r337639 = r337635 * r337638;
        double r337640 = r337639 * r337636;
        double r337641 = r337637 / r337640;
        return r337641;
}
↓
double f(double x, double y) {
        double r337642 = y;
        double r337643 = -5.819157750295796e-37;
        bool r337644 = r337642 <= r337643;
        double r337645 = 3.126349286155402e-87;
        bool r337646 = r337642 <= r337645;
        double r337647 = !r337646;
        bool r337648 = r337644 || r337647;
        double r337649 = 1.0;
        double r337650 = x;
        double r337651 = r337650 / r337642;
        double r337652 = r337649 + r337651;
        double r337653 = 2.0;
        double r337654 = r337652 / r337653;
        double r337655 = r337654 / r337650;
        double r337656 = r337649 / r337653;
        double r337657 = r337656 / r337642;
        double r337658 = r337650 + r337642;
        double r337659 = r337658 / r337650;
        double r337660 = r337657 * r337659;
        double r337661 = r337648 ? r337655 : r337660;
        return r337661;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.1
Target0.0
Herbie0.5
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.819157750295796e-37 or 3.126349286155402e-87 < y

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied associate-/r*0.8

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

    5. Simplified0.7

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

    6. Taylor expanded around 0 0.7

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

    if -5.819157750295796e-37 < y < 3.126349286155402e-87

    1. Initial program 18.0

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

    2. Simplified18.0

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

    4. Applied *-un-lft-identity18.0

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

    5. Applied times-frac0.1

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

    6. Simplified0.1

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

    7. Simplified0.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.819157750295796344692295718235260807105 \cdot 10^{-37} \lor \neg \left(y \le 3.126349286155401959304305196339847373255 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\frac{1 + \frac{x}{y}}{2}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{y} \cdot \frac{x + y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"

  :herbie-target
  (+ (/ 0.5 x) (/ 0.5 y))

  (/ (+ x y) (* (* x 2.0) y)))