Average Error: 15.4 → 0.3
Time: 4.0s
Precision: 64
\[\frac{x - y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.360330195182189619341025150081655268265 \cdot 10^{50} \lor \neg \left(x \le 2.245371265275845673758884064817345150736 \cdot 10^{45}\right):\\ \;\;\;\;\frac{\frac{x - y}{x \cdot 2}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{x - y}{2}}{y}\\ \end{array}\]
\frac{x - y}{\left(x \cdot 2\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.360330195182189619341025150081655268265 \cdot 10^{50} \lor \neg \left(x \le 2.245371265275845673758884064817345150736 \cdot 10^{45}\right):\\
\;\;\;\;\frac{\frac{x - y}{x \cdot 2}}{y}\\

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

\end{array}
double f(double x, double y) {
        double r487635 = x;
        double r487636 = y;
        double r487637 = r487635 - r487636;
        double r487638 = 2.0;
        double r487639 = r487635 * r487638;
        double r487640 = r487639 * r487636;
        double r487641 = r487637 / r487640;
        return r487641;
}
↓
double f(double x, double y) {
        double r487642 = x;
        double r487643 = -2.3603301951821896e+50;
        bool r487644 = r487642 <= r487643;
        double r487645 = 2.2453712652758457e+45;
        bool r487646 = r487642 <= r487645;
        double r487647 = !r487646;
        bool r487648 = r487644 || r487647;
        double r487649 = y;
        double r487650 = r487642 - r487649;
        double r487651 = 2.0;
        double r487652 = r487642 * r487651;
        double r487653 = r487650 / r487652;
        double r487654 = r487653 / r487649;
        double r487655 = 1.0;
        double r487656 = r487655 / r487642;
        double r487657 = r487650 / r487651;
        double r487658 = r487657 / r487649;
        double r487659 = r487656 * r487658;
        double r487660 = r487648 ? r487654 : r487659;
        return r487660;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.4
Target0.0
Herbie0.3
\[\frac{0.5}{y} - \frac{0.5}{x}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.3603301951821896e+50 or 2.2453712652758457e+45 < x

    1. Initial program 18.3

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

    3. Applied associate-/r*0.2

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

    if -2.3603301951821896e+50 < x < 2.2453712652758457e+45

    1. Initial program 13.2

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

    3. Applied associate-/r*14.8

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

    5. Applied *-un-lft-identity14.8

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

    6. Applied *-un-lft-identity14.8

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

    7. Applied times-frac14.7

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

    8. Applied times-frac0.4

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

    9. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.360330195182189619341025150081655268265 \cdot 10^{50} \lor \neg \left(x \le 2.245371265275845673758884064817345150736 \cdot 10^{45}\right):\\ \;\;\;\;\frac{\frac{x - y}{x \cdot 2}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{x - y}{2}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

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

  (/ (- x y) (* (* x 2) y)))