Average Error: 16.1 → 6.1
Time: 8.7m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 12778882773963.466796875:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\left(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}\right) \cdot \left(\left(\alpha + \beta\right) - 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 12778882773963.466796875:\\
\;\;\;\;\frac{\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(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}\right) \cdot \left(\left(\alpha + \beta\right) - 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r987931 = beta;
        double r987932 = alpha;
        double r987933 = r987931 - r987932;
        double r987934 = r987932 + r987931;
        double r987935 = 2.0;
        double r987936 = r987934 + r987935;
        double r987937 = r987933 / r987936;
        double r987938 = 1.0;
        double r987939 = r987937 + r987938;
        double r987940 = r987939 / r987935;
        return r987940;
}
↓
double f(double alpha, double beta) {
        double r987941 = alpha;
        double r987942 = 12778882773963.467;
        bool r987943 = r987941 <= r987942;
        double r987944 = beta;
        double r987945 = r987941 + r987944;
        double r987946 = 2.0;
        double r987947 = r987945 + r987946;
        double r987948 = r987941 / r987947;
        double r987949 = r987948 * r987948;
        double r987950 = 1.0;
        double r987951 = r987950 * r987950;
        double r987952 = r987948 * r987950;
        double r987953 = r987951 + r987952;
        double r987954 = r987949 + r987953;
        double r987955 = r987944 * r987954;
        double r987956 = 3.0;
        double r987957 = pow(r987948, r987956);
        double r987958 = pow(r987950, r987956);
        double r987959 = r987957 - r987958;
        double r987960 = r987947 * r987959;
        double r987961 = r987955 - r987960;
        double r987962 = r987950 + r987948;
        double r987963 = r987950 * r987962;
        double r987964 = r987945 * r987945;
        double r987965 = r987946 * r987946;
        double r987966 = r987964 - r987965;
        double r987967 = r987941 / r987966;
        double r987968 = r987948 * r987967;
        double r987969 = r987945 - r987946;
        double r987970 = r987968 * r987969;
        double r987971 = r987963 + r987970;
        double r987972 = r987971 * r987947;
        double r987973 = r987961 / r987972;
        double r987974 = r987973 / r987946;
        double r987975 = r987944 / r987947;
        double r987976 = 4.0;
        double r987977 = r987976 / r987941;
        double r987978 = r987977 / r987941;
        double r987979 = r987946 / r987941;
        double r987980 = 8.0;
        double r987981 = -r987980;
        double r987982 = pow(r987941, r987956);
        double r987983 = r987981 / r987982;
        double r987984 = r987979 - r987983;
        double r987985 = r987978 - r987984;
        double r987986 = r987975 - r987985;
        double r987987 = r987986 / r987946;
        double r987988 = r987943 ? r987974 : r987987;
        return r987988;
}

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 < 12778882773963.467

    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

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

    10. Applied flip-+0.3

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

    11. Applied associate-/r/0.3

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

    12. Applied associate-*r*0.3

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

    if 12778882773963.467 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    4. Applied associate-+l-48.5

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

    5. Taylor expanded around inf 18.4

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

    6. Simplified18.4

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 12778882773963.466796875:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\left(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}\right) \cdot \left(\left(\alpha + \beta\right) - 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))