Average Error: 16.3 → 6.8
Time: 4.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.20775003977996259 \cdot 10^{40}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.20775003977996259 \cdot 10^{40}:\\
\;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

\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 r101286 = beta;
        double r101287 = alpha;
        double r101288 = r101286 - r101287;
        double r101289 = r101287 + r101286;
        double r101290 = 2.0;
        double r101291 = r101289 + r101290;
        double r101292 = r101288 / r101291;
        double r101293 = 1.0;
        double r101294 = r101292 + r101293;
        double r101295 = r101294 / r101290;
        return r101295;
}
↓
double f(double alpha, double beta) {
        double r101296 = alpha;
        double r101297 = 1.2077500397799626e+40;
        bool r101298 = r101296 <= r101297;
        double r101299 = beta;
        double r101300 = 1.0;
        double r101301 = r101296 + r101299;
        double r101302 = 2.0;
        double r101303 = r101301 + r101302;
        double r101304 = r101300 / r101303;
        double r101305 = r101299 * r101304;
        double r101306 = r101296 / r101303;
        double r101307 = 1.0;
        double r101308 = r101306 - r101307;
        double r101309 = r101305 - r101308;
        double r101310 = r101309 / r101302;
        double r101311 = r101299 / r101303;
        double r101312 = 4.0;
        double r101313 = 2.0;
        double r101314 = pow(r101296, r101313);
        double r101315 = r101300 / r101314;
        double r101316 = r101312 * r101315;
        double r101317 = r101300 / r101296;
        double r101318 = r101302 * r101317;
        double r101319 = 8.0;
        double r101320 = 3.0;
        double r101321 = pow(r101296, r101320);
        double r101322 = r101300 / r101321;
        double r101323 = r101319 * r101322;
        double r101324 = r101318 + r101323;
        double r101325 = r101316 - r101324;
        double r101326 = r101311 - r101325;
        double r101327 = r101326 / r101302;
        double r101328 = r101298 ? r101310 : r101327;
        return r101328;
}

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 < 1.2077500397799626e+40

    1. Initial program 1.9

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

    3. Applied div-sub1.9

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

    4. Applied associate-+l-1.9

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

    6. Applied div-inv1.9

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

    if 1.2077500397799626e+40 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-48.9

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

    6. Applied flip--48.9

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

    7. Taylor expanded around inf 18.5

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.20775003977996259 \cdot 10^{40}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))