Average Error: 16.2 → 6.0
Time: 7.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2780207232289.98682:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2780207232289.98682:\\
\;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r120212 = beta;
        double r120213 = alpha;
        double r120214 = r120212 - r120213;
        double r120215 = r120213 + r120212;
        double r120216 = 2.0;
        double r120217 = r120215 + r120216;
        double r120218 = r120214 / r120217;
        double r120219 = 1.0;
        double r120220 = r120218 + r120219;
        double r120221 = r120220 / r120216;
        return r120221;
}
↓
double f(double alpha, double beta) {
        double r120222 = alpha;
        double r120223 = 2780207232289.987;
        bool r120224 = r120222 <= r120223;
        double r120225 = beta;
        double r120226 = r120222 + r120225;
        double r120227 = 2.0;
        double r120228 = r120226 + r120227;
        double r120229 = r120225 / r120228;
        double r120230 = r120222 / r120228;
        double r120231 = 1.0;
        double r120232 = r120230 - r120231;
        double r120233 = r120229 - r120232;
        double r120234 = r120233 / r120227;
        double r120235 = log(r120234);
        double r120236 = exp(r120235);
        double r120237 = 4.0;
        double r120238 = 1.0;
        double r120239 = 2.0;
        double r120240 = pow(r120222, r120239);
        double r120241 = r120238 / r120240;
        double r120242 = r120237 * r120241;
        double r120243 = r120238 / r120222;
        double r120244 = r120227 * r120243;
        double r120245 = 8.0;
        double r120246 = 3.0;
        double r120247 = pow(r120222, r120246);
        double r120248 = r120238 / r120247;
        double r120249 = r120245 * r120248;
        double r120250 = r120244 + r120249;
        double r120251 = r120242 - r120250;
        double r120252 = r120229 - r120251;
        double r120253 = r120252 / r120227;
        double r120254 = r120224 ? r120236 : r120253;
        return r120254;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2780207232289.987

    1. Initial program 0.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub0.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-0.3

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-exp-log0.3

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{\color{blue}{e^{\log 2}}}\]

    7. Applied add-exp-log0.3

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}}{e^{\log 2}}\]

    8. Applied div-exp0.3

      \[\leadsto \color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) - \log 2}}\]

    9. Simplified0.3

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)}}\]

    if 2780207232289.987 < alpha

    1. Initial program 50.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub50.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-48.7

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]

    5. Taylor expanded around inf 18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2780207232289.98682:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))