Average Error: 16.0 → 6.1
Time: 29.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 246471310145705.31:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)} \cdot \frac{{\alpha}^{3}}{\left(\alpha + \beta\right) + 2} - {1}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 246471310145705.31:\\
\;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)} \cdot \frac{{\alpha}^{3}}{\left(\alpha + \beta\right) + 2} - {1}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r133986 = beta;
        double r133987 = alpha;
        double r133988 = r133986 - r133987;
        double r133989 = r133987 + r133986;
        double r133990 = 2.0;
        double r133991 = r133989 + r133990;
        double r133992 = r133988 / r133991;
        double r133993 = 1.0;
        double r133994 = r133992 + r133993;
        double r133995 = r133994 / r133990;
        return r133995;
}
↓
double f(double alpha, double beta) {
        double r133996 = alpha;
        double r133997 = 246471310145705.3;
        bool r133998 = r133996 <= r133997;
        double r133999 = beta;
        double r134000 = r133996 + r133999;
        double r134001 = 2.0;
        double r134002 = r134000 + r134001;
        double r134003 = r133996 / r134002;
        double r134004 = r134003 * r134003;
        double r134005 = 1.0;
        double r134006 = r134003 + r134005;
        double r134007 = r134005 * r134006;
        double r134008 = r134004 + r134007;
        double r134009 = r133999 * r134008;
        double r134010 = 1.0;
        double r134011 = r134002 * r134002;
        double r134012 = r134010 / r134011;
        double r134013 = 3.0;
        double r134014 = pow(r133996, r134013);
        double r134015 = r134014 / r134002;
        double r134016 = r134012 * r134015;
        double r134017 = pow(r134005, r134013);
        double r134018 = r134016 - r134017;
        double r134019 = r134002 * r134018;
        double r134020 = r134009 - r134019;
        double r134021 = r134002 * r134008;
        double r134022 = r134020 / r134021;
        double r134023 = r134022 / r134001;
        double r134024 = r133999 / r134002;
        double r134025 = 4.0;
        double r134026 = r133996 * r133996;
        double r134027 = r134025 / r134026;
        double r134028 = 8.0;
        double r134029 = r134028 / r134014;
        double r134030 = r134001 / r133996;
        double r134031 = r134029 + r134030;
        double r134032 = r134027 - r134031;
        double r134033 = r134024 - r134032;
        double r134034 = r134033 / r134001;
        double r134035 = r133998 ? r134023 : r134034;
        return r134035;
}

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 < 246471310145705.3

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    6. Applied flip3--0.3

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

    7. Applied frac-sub0.3

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

    8. Simplified0.3

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

    9. Simplified0.3

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}}}{2}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt0.4

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

    12. Applied *-un-lft-identity0.4

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

    13. Applied times-frac0.4

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

    14. Applied unpow-prod-down0.4

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

    15. Simplified0.4

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

    16. Simplified0.3

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

    if 246471310145705.3 < 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-48.4

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

    5. Taylor expanded around inf 18.6

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

    6. Simplified18.6

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 246471310145705.31:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)} \cdot \frac{{\alpha}^{3}}{\left(\alpha + \beta\right) + 2} - {1}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))