Average Error: 16.4 → 6.7
Time: 20.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3427632623736984.5:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{{\left(2.0 + \left(\beta + \alpha\right)\right)}^{\frac{1}{3}} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta}}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sqrt[3]{2.0 + \left(\beta + \alpha\right)}\right)\right)\right)\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3427632623736984.5:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{{\left(2.0 + \left(\beta + \alpha\right)\right)}^{\frac{1}{3}} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta}}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sqrt[3]{2.0 + \left(\beta + \alpha\right)}\right)\right)\right)\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r1342082 = beta;
        double r1342083 = alpha;
        double r1342084 = r1342082 - r1342083;
        double r1342085 = r1342083 + r1342082;
        double r1342086 = 2.0;
        double r1342087 = r1342085 + r1342086;
        double r1342088 = r1342084 / r1342087;
        double r1342089 = 1.0;
        double r1342090 = r1342088 + r1342089;
        double r1342091 = r1342090 / r1342086;
        return r1342091;
}
↓
double f(double alpha, double beta) {
        double r1342092 = alpha;
        double r1342093 = 3427632623736984.5;
        bool r1342094 = r1342092 <= r1342093;
        double r1342095 = beta;
        double r1342096 = cbrt(r1342095);
        double r1342097 = r1342096 * r1342096;
        double r1342098 = 2.0;
        double r1342099 = r1342095 + r1342092;
        double r1342100 = r1342098 + r1342099;
        double r1342101 = 0.3333333333333333;
        double r1342102 = pow(r1342100, r1342101);
        double r1342103 = cbrt(r1342100);
        double r1342104 = r1342102 * r1342103;
        double r1342105 = r1342097 / r1342104;
        double r1342106 = log1p(r1342103);
        double r1342107 = expm1(r1342106);
        double r1342108 = r1342096 / r1342107;
        double r1342109 = r1342105 * r1342108;
        double r1342110 = r1342092 / r1342100;
        double r1342111 = 1.0;
        double r1342112 = r1342110 - r1342111;
        double r1342113 = r1342109 - r1342112;
        double r1342114 = r1342113 / r1342098;
        double r1342115 = r1342095 / r1342100;
        double r1342116 = 1.0;
        double r1342117 = r1342092 * r1342092;
        double r1342118 = r1342116 / r1342117;
        double r1342119 = 4.0;
        double r1342120 = 8.0;
        double r1342121 = r1342120 / r1342092;
        double r1342122 = r1342119 - r1342121;
        double r1342123 = r1342118 * r1342122;
        double r1342124 = r1342098 / r1342092;
        double r1342125 = r1342123 - r1342124;
        double r1342126 = r1342115 - r1342125;
        double r1342127 = r1342126 / r1342098;
        double r1342128 = r1342094 ? r1342114 : r1342127;
        return r1342128;
}

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 < 3427632623736984.5

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    6. Applied add-cube-cbrt0.6

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

    7. Applied add-cube-cbrt0.4

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

    8. Applied times-frac0.4

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

    10. Applied expm1-log1p-u1.5

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right)\right)\right)\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
    11. Using strategy rm

    12. Applied pow1/31.5

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \color{blue}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{\frac{1}{3}}}} \cdot \frac{\sqrt[3]{\beta}}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right)\right)\right)\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    if 3427632623736984.5 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.8

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

    5. Taylor expanded around -inf 17.8

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

    6. Simplified17.8

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3427632623736984.5:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{{\left(2.0 + \left(\beta + \alpha\right)\right)}^{\frac{1}{3}} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta}}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sqrt[3]{2.0 + \left(\beta + \alpha\right)}\right)\right)\right)\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +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))