Average Error: 15.2 → 0.2
Time: 9.5s
Precision: 64
\[\frac{x - y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.988741544918683514691243360830679767552 \cdot 10^{-57} \lor \neg \left(x \le 2.652081687524138675796576957148481763096 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\frac{x - y}{x \cdot 2}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{x - y}{2}}{y}\\ \end{array}\]
\frac{x - y}{\left(x \cdot 2\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.988741544918683514691243360830679767552 \cdot 10^{-57} \lor \neg \left(x \le 2.652081687524138675796576957148481763096 \cdot 10^{-25}\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 r307622 = x;
        double r307623 = y;
        double r307624 = r307622 - r307623;
        double r307625 = 2.0;
        double r307626 = r307622 * r307625;
        double r307627 = r307626 * r307623;
        double r307628 = r307624 / r307627;
        return r307628;
}
↓
double f(double x, double y) {
        double r307629 = x;
        double r307630 = -3.9887415449186835e-57;
        bool r307631 = r307629 <= r307630;
        double r307632 = 2.6520816875241387e-25;
        bool r307633 = r307629 <= r307632;
        double r307634 = !r307633;
        bool r307635 = r307631 || r307634;
        double r307636 = y;
        double r307637 = r307629 - r307636;
        double r307638 = 2.0;
        double r307639 = r307629 * r307638;
        double r307640 = r307637 / r307639;
        double r307641 = r307640 / r307636;
        double r307642 = 1.0;
        double r307643 = r307642 / r307629;
        double r307644 = r307637 / r307638;
        double r307645 = r307644 / r307636;
        double r307646 = r307643 * r307645;
        double r307647 = r307635 ? r307641 : r307646;
        return r307647;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.2
Target0.0
Herbie0.2
\[\frac{0.5}{y} - \frac{0.5}{x}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.9887415449186835e-57 or 2.6520816875241387e-25 < x

    1. Initial program 13.8

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

    3. Applied associate-/r*0.3

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

    if -3.9887415449186835e-57 < x < 2.6520816875241387e-25

    1. Initial program 17.0

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

    3. Applied associate-/r*18.1

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

    5. Applied *-un-lft-identity18.1

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

    6. Applied *-un-lft-identity18.1

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

    7. Applied times-frac18.1

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

    8. Applied times-frac0.1

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

    9. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.988741544918683514691243360830679767552 \cdot 10^{-57} \lor \neg \left(x \le 2.652081687524138675796576957148481763096 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\frac{x - y}{x \cdot 2}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{x - y}{2}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))