Average Error: 15.9 → 5.9
Time: 22.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 94141546.97903764247894287109375:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 94141546.97903764247894287109375:\\
\;\;\;\;\frac{{e}^{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r116205 = beta;
        double r116206 = alpha;
        double r116207 = r116205 - r116206;
        double r116208 = r116206 + r116205;
        double r116209 = 2.0;
        double r116210 = r116208 + r116209;
        double r116211 = r116207 / r116210;
        double r116212 = 1.0;
        double r116213 = r116211 + r116212;
        double r116214 = r116213 / r116209;
        return r116214;
}
↓
double f(double alpha, double beta) {
        double r116215 = alpha;
        double r116216 = 94141546.97903764;
        bool r116217 = r116215 <= r116216;
        double r116218 = exp(1.0);
        double r116219 = beta;
        double r116220 = r116215 + r116219;
        double r116221 = 2.0;
        double r116222 = r116220 + r116221;
        double r116223 = r116219 / r116222;
        double r116224 = r116215 / r116222;
        double r116225 = 1.0;
        double r116226 = r116224 - r116225;
        double r116227 = r116223 - r116226;
        double r116228 = log(r116227);
        double r116229 = pow(r116218, r116228);
        double r116230 = r116229 / r116221;
        double r116231 = 4.0;
        double r116232 = r116215 * r116215;
        double r116233 = r116231 / r116232;
        double r116234 = r116221 / r116215;
        double r116235 = 8.0;
        double r116236 = 3.0;
        double r116237 = pow(r116215, r116236);
        double r116238 = r116235 / r116237;
        double r116239 = r116234 + r116238;
        double r116240 = r116233 - r116239;
        double r116241 = r116223 - r116240;
        double r116242 = r116241 / r116221;
        double r116243 = r116217 ? r116230 : r116242;
        return r116243;
}

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 < 94141546.97903764

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-exp-log0.1

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

    8. Applied pow10.1

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

    9. Applied log-pow0.1

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

    10. Applied exp-prod0.4

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

    11. Simplified0.4

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

    if 94141546.97903764 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    4. Applied associate-+l-48.6

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

    5. Taylor expanded around inf 17.7

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

    6. Simplified17.7

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 94141546.97903764247894287109375:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))