Average Error: 16.4 → 6.3
Time: 15.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 76510312.3131744563579559326171875:\\ \;\;\;\;\frac{e^{\log \left(1 + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + 2\right) + \beta} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 76510312.3131744563579559326171875:\\
\;\;\;\;\frac{e^{\log \left(1 + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r77215 = beta;
        double r77216 = alpha;
        double r77217 = r77215 - r77216;
        double r77218 = r77216 + r77215;
        double r77219 = 2.0;
        double r77220 = r77218 + r77219;
        double r77221 = r77217 / r77220;
        double r77222 = 1.0;
        double r77223 = r77221 + r77222;
        double r77224 = r77223 / r77219;
        return r77224;
}
↓
double f(double alpha, double beta) {
        double r77225 = alpha;
        double r77226 = 76510312.31317446;
        bool r77227 = r77225 <= r77226;
        double r77228 = 1.0;
        double r77229 = beta;
        double r77230 = r77229 - r77225;
        double r77231 = 2.0;
        double r77232 = r77225 + r77231;
        double r77233 = r77232 + r77229;
        double r77234 = r77230 / r77233;
        double r77235 = r77228 + r77234;
        double r77236 = log(r77235);
        double r77237 = exp(r77236);
        double r77238 = r77237 / r77231;
        double r77239 = r77229 / r77233;
        double r77240 = 4.0;
        double r77241 = r77225 * r77225;
        double r77242 = r77240 / r77241;
        double r77243 = 8.0;
        double r77244 = 3.0;
        double r77245 = pow(r77225, r77244);
        double r77246 = r77243 / r77245;
        double r77247 = r77231 / r77225;
        double r77248 = r77246 + r77247;
        double r77249 = r77242 - r77248;
        double r77250 = r77239 - r77249;
        double r77251 = r77250 / r77231;
        double r77252 = r77227 ? r77238 : r77251;
        return r77252;
}

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 < 76510312.31317446

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-exp-log0.1

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

    5. Simplified0.1

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

    if 76510312.31317446 < alpha

    1. Initial program 49.6

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

    2. Simplified49.6

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

    4. Applied div-sub49.6

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

    5. Applied associate-+l-48.0

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

    6. Simplified48.0

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

    7. Taylor expanded around inf 19.0

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

    8. Simplified19.0

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 76510312.3131744563579559326171875:\\ \;\;\;\;\frac{e^{\log \left(1 + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + 2\right) + \beta} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))