Average Error: 15.3 → 0.6
Time: 10.8s
Precision: 64
\[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -120878309475402.03125 \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -5.545286165337571374349144190718927055493 \cdot 10^{-303} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0\right) \land \frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 8.162914331366078273047657170137319920997 \cdot 10^{-36}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-y}{-\left(x - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \end{array}\]
\frac{\left(x \cdot 2\right) \cdot y}{x - y}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -120878309475402.03125 \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -5.545286165337571374349144190718927055493 \cdot 10^{-303} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0\right) \land \frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 8.162914331366078273047657170137319920997 \cdot 10^{-36}\right):\\
\;\;\;\;\left(x \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-y}{-\left(x - y\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\

\end{array}
double f(double x, double y) {
        double r364752 = x;
        double r364753 = 2.0;
        double r364754 = r364752 * r364753;
        double r364755 = y;
        double r364756 = r364754 * r364755;
        double r364757 = r364752 - r364755;
        double r364758 = r364756 / r364757;
        return r364758;
}
↓
double f(double x, double y) {
        double r364759 = x;
        double r364760 = 2.0;
        double r364761 = r364759 * r364760;
        double r364762 = y;
        double r364763 = r364761 * r364762;
        double r364764 = r364759 - r364762;
        double r364765 = r364763 / r364764;
        double r364766 = -120878309475402.03;
        bool r364767 = r364765 <= r364766;
        double r364768 = -5.5452861653375714e-303;
        bool r364769 = r364765 <= r364768;
        double r364770 = 0.0;
        bool r364771 = r364765 <= r364770;
        double r364772 = !r364771;
        double r364773 = 8.162914331366078e-36;
        bool r364774 = r364765 <= r364773;
        bool r364775 = r364772 && r364774;
        bool r364776 = r364769 || r364775;
        double r364777 = !r364776;
        bool r364778 = r364767 || r364777;
        double r364779 = -r364762;
        double r364780 = -r364764;
        double r364781 = r364779 / r364780;
        double r364782 = expm1(r364781);
        double r364783 = log1p(r364782);
        double r364784 = r364761 * r364783;
        double r364785 = r364778 ? r364784 : r364765;
        return r364785;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -1.721044263414944729490876394165887012892 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x \lt 83645045635564432:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (* x 2.0) y) (- x y)) < -120878309475402.03 or -5.5452861653375714e-303 < (/ (* (* x 2.0) y) (- x y)) < 0.0 or 8.162914331366078e-36 < (/ (* (* x 2.0) y) (- x y))

    1. Initial program 34.9

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

    3. Applied *-un-lft-identity34.9

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

    4. Applied times-frac0.6

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

    5. Simplified0.6

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

    7. Applied frac-2neg0.6

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

    9. Applied log1p-expm1-u0.7

      \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-y}{-\left(x - y\right)}\right)\right)}\]

    if -120878309475402.03 < (/ (* (* x 2.0) y) (- x y)) < -5.5452861653375714e-303 or 0.0 < (/ (* (* x 2.0) y) (- x y)) < 8.162914331366078e-36

    1. Initial program 5.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -120878309475402.03125 \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -5.545286165337571374349144190718927055493 \cdot 10^{-303} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0\right) \land \frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 8.162914331366078273047657170137319920997 \cdot 10^{-36}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-y}{-\left(x - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564432) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y)))

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