Average Error: 16.0 → 6.0
Time: 6.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 45468567748.04238128662109375:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1}}{\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}} \cdot \frac{\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1}}{\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 45468567748.04238128662109375:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1}}{\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}} \cdot \frac{\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1}}{\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r91994 = beta;
        double r91995 = alpha;
        double r91996 = r91994 - r91995;
        double r91997 = r91995 + r91994;
        double r91998 = 2.0;
        double r91999 = r91997 + r91998;
        double r92000 = r91996 / r91999;
        double r92001 = 1.0;
        double r92002 = r92000 + r92001;
        double r92003 = r92002 / r91998;
        return r92003;
}
↓
double f(double alpha, double beta) {
        double r92004 = alpha;
        double r92005 = 45468567748.04238;
        bool r92006 = r92004 <= r92005;
        double r92007 = beta;
        double r92008 = r92004 + r92007;
        double r92009 = 2.0;
        double r92010 = r92008 + r92009;
        double r92011 = r92007 / r92010;
        double r92012 = r92004 / r92010;
        double r92013 = r92012 * r92012;
        double r92014 = 1.0;
        double r92015 = r92014 * r92014;
        double r92016 = r92013 - r92015;
        double r92017 = cbrt(r92016);
        double r92018 = r92012 + r92014;
        double r92019 = sqrt(r92018);
        double r92020 = r92017 / r92019;
        double r92021 = r92017 * r92017;
        double r92022 = r92021 / r92019;
        double r92023 = r92020 * r92022;
        double r92024 = r92011 - r92023;
        double r92025 = r92024 / r92009;
        double r92026 = 4.0;
        double r92027 = r92026 / r92004;
        double r92028 = r92027 / r92004;
        double r92029 = 8.0;
        double r92030 = -r92029;
        double r92031 = 3.0;
        double r92032 = pow(r92004, r92031);
        double r92033 = r92030 / r92032;
        double r92034 = r92028 + r92033;
        double r92035 = -r92009;
        double r92036 = r92035 / r92004;
        double r92037 = r92034 + r92036;
        double r92038 = r92011 - r92037;
        double r92039 = r92038 / r92009;
        double r92040 = r92006 ? r92025 : r92039;
        return r92040;
}

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 < 45468567748.04238

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied add-log-exp0.2

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

    7. Applied add-log-exp0.2

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

    8. Applied diff-log0.2

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

    9. Simplified0.2

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

    11. Applied flip--0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\color{blue}{\frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}}\right)}{2}\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt0.2

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

    14. Applied add-cube-cbrt0.2

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

    15. Applied times-frac0.2

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

    16. Applied exp-prod0.2

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

    17. Applied log-pow0.2

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

    18. Simplified0.2

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

    if 45468567748.04238 < alpha

    1. Initial program 49.8

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

    3. Applied div-sub49.8

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

    4. Applied associate-+l-48.2

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

    5. Taylor expanded around inf 18.3

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

    6. Simplified18.3

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 45468567748.04238128662109375:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1}}{\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}} \cdot \frac{\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot 1}}{\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))