Average Error: 16.2 → 5.9
Time: 4.0s
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 r105068 = beta;
        double r105069 = alpha;
        double r105070 = r105068 - r105069;
        double r105071 = r105069 + r105068;
        double r105072 = 2.0;
        double r105073 = r105071 + r105072;
        double r105074 = r105070 / r105073;
        double r105075 = 1.0;
        double r105076 = r105074 + r105075;
        double r105077 = r105076 / r105072;
        return r105077;
}
↓
double f(double alpha, double beta) {
        double r105078 = alpha;
        double r105079 = 92108.89598052698;
        bool r105080 = r105078 <= r105079;
        double r105081 = beta;
        double r105082 = r105078 + r105081;
        double r105083 = 2.0;
        double r105084 = r105082 + r105083;
        double r105085 = r105081 / r105084;
        double r105086 = 3.0;
        double r105087 = pow(r105085, r105086);
        double r105088 = cbrt(r105087);
        double r105089 = r105078 / r105084;
        double r105090 = 1.0;
        double r105091 = r105089 - r105090;
        double r105092 = r105088 - r105091;
        double r105093 = r105092 / r105083;
        double r105094 = 1.0;
        double r105095 = r105084 / r105081;
        double r105096 = r105094 / r105095;
        double r105097 = 4.0;
        double r105098 = 2.0;
        double r105099 = pow(r105078, r105098);
        double r105100 = r105094 / r105099;
        double r105101 = r105097 * r105100;
        double r105102 = r105094 / r105078;
        double r105103 = r105083 * r105102;
        double r105104 = 8.0;
        double r105105 = pow(r105078, r105086);
        double r105106 = r105094 / r105105;
        double r105107 = r105104 * r105106;
        double r105108 = r105103 + r105107;
        double r105109 = r105101 - r105108;
        double r105110 = r105096 - r105109;
        double r105111 = r105110 / r105083;
        double r105112 = r105080 ? r105093 : r105111;
        return r105112;
}

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))