Average Error: 16.1 → 6.0
Time: 7.3m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 27243826751.57198:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}}\right) \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} - \frac{\sqrt[3]{\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left(\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)\right)}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 27243826751.57198:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}}\right) \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} - \frac{\sqrt[3]{\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left(\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)\right)}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right)}}{2.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r30448132 = beta;
        double r30448133 = alpha;
        double r30448134 = r30448132 - r30448133;
        double r30448135 = r30448133 + r30448132;
        double r30448136 = 2.0;
        double r30448137 = r30448135 + r30448136;
        double r30448138 = r30448134 / r30448137;
        double r30448139 = 1.0;
        double r30448140 = r30448138 + r30448139;
        double r30448141 = r30448140 / r30448136;
        return r30448141;
}
↓
double f(double alpha, double beta) {
        double r30448142 = alpha;
        double r30448143 = 27243826751.57198;
        bool r30448144 = r30448142 <= r30448143;
        double r30448145 = beta;
        double r30448146 = 2.0;
        double r30448147 = r30448145 + r30448142;
        double r30448148 = r30448146 + r30448147;
        double r30448149 = r30448145 / r30448148;
        double r30448150 = cbrt(r30448149);
        double r30448151 = r30448150 * r30448150;
        double r30448152 = r30448151 * r30448150;
        double r30448153 = r30448142 / r30448148;
        double r30448154 = 3.0;
        double r30448155 = pow(r30448153, r30448154);
        double r30448156 = 1.0;
        double r30448157 = pow(r30448156, r30448154);
        double r30448158 = r30448155 - r30448157;
        double r30448159 = r30448158 * r30448158;
        double r30448160 = r30448158 * r30448159;
        double r30448161 = cbrt(r30448160);
        double r30448162 = r30448153 * r30448153;
        double r30448163 = r30448156 * r30448153;
        double r30448164 = r30448156 * r30448156;
        double r30448165 = r30448163 + r30448164;
        double r30448166 = r30448162 + r30448165;
        double r30448167 = r30448161 / r30448166;
        double r30448168 = r30448152 - r30448167;
        double r30448169 = r30448168 / r30448146;
        double r30448170 = 4.0;
        double r30448171 = r30448142 * r30448142;
        double r30448172 = r30448170 / r30448171;
        double r30448173 = 8.0;
        double r30448174 = r30448173 / r30448171;
        double r30448175 = r30448174 / r30448142;
        double r30448176 = r30448146 / r30448142;
        double r30448177 = r30448175 + r30448176;
        double r30448178 = r30448172 - r30448177;
        double r30448179 = r30448149 - r30448178;
        double r30448180 = r30448179 / r30448146;
        double r30448181 = r30448144 ? r30448169 : r30448180;
        return r30448181;
}

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 < 27243826751.57198

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
    7. Using strategy rm

    8. Applied flip3--0.2

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]
    9. Using strategy rm

    10. Applied add-cbrt-cube0.2

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \frac{\color{blue}{\sqrt[3]{\left(\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]

    if 27243826751.57198 < alpha

    1. Initial program 49.4

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

    3. Applied div-sub49.3

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

    4. Applied associate-+l-47.9

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

    5. Taylor expanded around inf 18.1

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

    6. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 27243826751.57198:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}}\right) \cdot \sqrt[3]{\frac{\beta}{2.0 + \left(\beta + \alpha\right)}} - \frac{\sqrt[3]{\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left(\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)\right)}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +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))