Average Error: 16.3 → 6.5
Time: 1.5m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.533190092860505 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{1}{\sqrt{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.533190092860505 \cdot 10^{+28}:\\
\;\;\;\;\frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{1}{\sqrt{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3749304 = beta;
        double r3749305 = alpha;
        double r3749306 = r3749304 - r3749305;
        double r3749307 = r3749305 + r3749304;
        double r3749308 = 2.0;
        double r3749309 = r3749307 + r3749308;
        double r3749310 = r3749306 / r3749309;
        double r3749311 = 1.0;
        double r3749312 = r3749310 + r3749311;
        double r3749313 = r3749312 / r3749308;
        return r3749313;
}
↓
double f(double alpha, double beta) {
        double r3749314 = alpha;
        double r3749315 = 3.533190092860505e+28;
        bool r3749316 = r3749314 <= r3749315;
        double r3749317 = beta;
        double r3749318 = r3749317 - r3749314;
        double r3749319 = r3749314 + r3749317;
        double r3749320 = 2.0;
        double r3749321 = r3749319 + r3749320;
        double r3749322 = sqrt(r3749321);
        double r3749323 = r3749318 / r3749322;
        double r3749324 = 1.0;
        double r3749325 = r3749324 / r3749322;
        double r3749326 = r3749323 * r3749325;
        double r3749327 = 1.0;
        double r3749328 = r3749326 + r3749327;
        double r3749329 = r3749328 / r3749320;
        double r3749330 = r3749317 / r3749321;
        double r3749331 = 4.0;
        double r3749332 = r3749314 * r3749314;
        double r3749333 = r3749331 / r3749332;
        double r3749334 = r3749320 / r3749314;
        double r3749335 = 8.0;
        double r3749336 = r3749335 / r3749332;
        double r3749337 = r3749336 / r3749314;
        double r3749338 = r3749334 + r3749337;
        double r3749339 = r3749333 - r3749338;
        double r3749340 = r3749330 - r3749339;
        double r3749341 = r3749340 / r3749320;
        double r3749342 = r3749316 ? r3749329 : r3749341;
        return r3749342;
}

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 < 3.533190092860505e+28

    1. Initial program 1.2

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt1.3

      \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt{\left(\alpha + \beta\right) + 2.0}}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity1.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\beta - \alpha\right)}}{\sqrt{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\]

    5. Applied times-frac1.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2.0}}} + 1.0}{2.0}\]

    if 3.533190092860505e+28 < alpha

    1. Initial program 50.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub50.8

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-49.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]

    5. Taylor expanded around inf 18.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    6. Simplified18.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} + \frac{2.0}{\alpha}\right)\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.533190092860505 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{1}{\sqrt{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))