Average Error: 16.2 → 5.9
Time: 4.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 92108.895980526984:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 92108.895980526984:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \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 r98203 = beta;
        double r98204 = alpha;
        double r98205 = r98203 - r98204;
        double r98206 = r98204 + r98203;
        double r98207 = 2.0;
        double r98208 = r98206 + r98207;
        double r98209 = r98205 / r98208;
        double r98210 = 1.0;
        double r98211 = r98209 + r98210;
        double r98212 = r98211 / r98207;
        return r98212;
}
↓
double f(double alpha, double beta) {
        double r98213 = alpha;
        double r98214 = 92108.89598052698;
        bool r98215 = r98213 <= r98214;
        double r98216 = beta;
        double r98217 = r98213 + r98216;
        double r98218 = 2.0;
        double r98219 = r98217 + r98218;
        double r98220 = r98216 / r98219;
        double r98221 = 3.0;
        double r98222 = pow(r98220, r98221);
        double r98223 = cbrt(r98222);
        double r98224 = r98213 / r98219;
        double r98225 = 1.0;
        double r98226 = r98224 - r98225;
        double r98227 = r98223 - r98226;
        double r98228 = r98227 / r98218;
        double r98229 = 1.0;
        double r98230 = r98219 / r98216;
        double r98231 = r98229 / r98230;
        double r98232 = 4.0;
        double r98233 = 2.0;
        double r98234 = pow(r98213, r98233);
        double r98235 = r98229 / r98234;
        double r98236 = r98232 * r98235;
        double r98237 = r98229 / r98213;
        double r98238 = r98218 * r98237;
        double r98239 = 8.0;
        double r98240 = pow(r98213, r98221);
        double r98241 = r98229 / r98240;
        double r98242 = r98239 * r98241;
        double r98243 = r98238 + r98242;
        double r98244 = r98236 - r98243;
        double r98245 = r98231 - r98244;
        double r98246 = r98245 / r98218;
        double r98247 = r98215 ? r98228 : r98246;
        return r98247;
}

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 < 92108.89598052698

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    4. Applied associate-+l-0.0

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

    6. Applied add-cbrt-cube11.7

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

    7. Applied add-cbrt-cube14.4

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

    8. Applied cbrt-undiv14.4

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

    9. Simplified0.0

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

    if 92108.89598052698 < 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-48.3

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

    6. Applied clear-num48.3

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

    7. Taylor expanded around inf 18.3

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 92108.895980526984:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))