Average Error: 16.2 → 5.8
Time: 9.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1773850741.520723819732666015625:\\ \;\;\;\;\frac{1 \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1773850741.520723819732666015625:\\
\;\;\;\;\frac{1 \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r66240 = beta;
        double r66241 = alpha;
        double r66242 = r66240 - r66241;
        double r66243 = r66241 + r66240;
        double r66244 = 2.0;
        double r66245 = r66243 + r66244;
        double r66246 = r66242 / r66245;
        double r66247 = 1.0;
        double r66248 = r66246 + r66247;
        double r66249 = r66248 / r66244;
        return r66249;
}
↓
double f(double alpha, double beta) {
        double r66250 = alpha;
        double r66251 = 1773850741.5207238;
        bool r66252 = r66250 <= r66251;
        double r66253 = 1.0;
        double r66254 = beta;
        double r66255 = r66250 + r66254;
        double r66256 = 2.0;
        double r66257 = r66255 + r66256;
        double r66258 = r66257 / r66254;
        double r66259 = r66253 / r66258;
        double r66260 = r66253 * r66259;
        double r66261 = r66250 / r66257;
        double r66262 = 1.0;
        double r66263 = r66261 - r66262;
        double r66264 = r66260 - r66263;
        double r66265 = r66264 / r66256;
        double r66266 = sqrt(r66257);
        double r66267 = r66253 / r66266;
        double r66268 = r66254 / r66266;
        double r66269 = r66267 * r66268;
        double r66270 = 4.0;
        double r66271 = r66270 / r66250;
        double r66272 = r66271 / r66250;
        double r66273 = 8.0;
        double r66274 = -r66273;
        double r66275 = 3.0;
        double r66276 = pow(r66250, r66275);
        double r66277 = r66274 / r66276;
        double r66278 = r66272 + r66277;
        double r66279 = -r66256;
        double r66280 = r66279 / r66250;
        double r66281 = r66278 + r66280;
        double r66282 = r66269 - r66281;
        double r66283 = r66282 / r66256;
        double r66284 = r66252 ? r66265 : r66283;
        return r66284;
}

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 < 1773850741.5207238

    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-sqr-sqrt0.1

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

    7. Applied *-un-lft-identity0.1

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

    8. Applied times-frac0.2

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

    10. Applied *-un-lft-identity0.2

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

    11. Applied associate-*l*0.2

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

    12. Simplified0.1

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

    14. Applied clear-num0.1

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

    if 1773850741.5207238 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub50.0

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

    4. Applied associate-+l-48.5

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

    6. Applied add-sqr-sqrt48.5

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

    7. Applied *-un-lft-identity48.5

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

    8. Applied times-frac48.5

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

    9. Taylor expanded around inf 17.9

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

    10. Simplified17.9

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1773850741.520723819732666015625:\\ \;\;\;\;\frac{1 \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))