Average Error: 15.3 → 0.6
Time: 14.3s
Precision: 64
\[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -120878309475402.03125 \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -5.545286165337571374349144190718927055493 \cdot 10^{-303} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0\right) \land \frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 8.162914331366078273047657170137319920997 \cdot 10^{-36}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{x - y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \end{array}\]
\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}{x - y}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r363885 = x;
        double r363886 = 2.0;
        double r363887 = r363885 * r363886;
        double r363888 = y;
        double r363889 = r363887 * r363888;
        double r363890 = r363885 - r363888;
        double r363891 = r363889 / r363890;
        return r363891;
}
↓
double f(double x, double y) {
        double r363892 = x;
        double r363893 = 2.0;
        double r363894 = r363892 * r363893;
        double r363895 = y;
        double r363896 = r363894 * r363895;
        double r363897 = r363892 - r363895;
        double r363898 = r363896 / r363897;
        double r363899 = -120878309475402.03;
        bool r363900 = r363898 <= r363899;
        double r363901 = -5.5452861653375714e-303;
        bool r363902 = r363898 <= r363901;
        double r363903 = 0.0;
        bool r363904 = r363898 <= r363903;
        double r363905 = !r363904;
        double r363906 = 8.162914331366078e-36;
        bool r363907 = r363898 <= r363906;
        bool r363908 = r363905 && r363907;
        bool r363909 = r363902 || r363908;
        double r363910 = !r363909;
        bool r363911 = r363900 || r363910;
        double r363912 = r363895 / r363897;
        double r363913 = expm1(r363912);
        double r363914 = log1p(r363913);
        double r363915 = r363894 * r363914;
        double r363916 = r363911 ? r363915 : r363898;
        return r363916;
}

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 log1p-expm1-u0.7

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

    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

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

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)))