Average Error: 16.1 → 6.1
Time: 22.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 10272132967178968:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 10272132967178968:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r4531918 = beta;
        double r4531919 = alpha;
        double r4531920 = r4531918 - r4531919;
        double r4531921 = r4531919 + r4531918;
        double r4531922 = 2.0;
        double r4531923 = r4531921 + r4531922;
        double r4531924 = r4531920 / r4531923;
        double r4531925 = 1.0;
        double r4531926 = r4531924 + r4531925;
        double r4531927 = r4531926 / r4531922;
        return r4531927;
}
↓
double f(double alpha, double beta) {
        double r4531928 = alpha;
        double r4531929 = 10272132967178968.0;
        bool r4531930 = r4531928 <= r4531929;
        double r4531931 = beta;
        double r4531932 = 2.0;
        double r4531933 = r4531931 + r4531928;
        double r4531934 = r4531932 + r4531933;
        double r4531935 = r4531931 / r4531934;
        double r4531936 = r4531935 * r4531935;
        double r4531937 = r4531935 * r4531936;
        double r4531938 = cbrt(r4531937);
        double r4531939 = r4531928 / r4531934;
        double r4531940 = 1.0;
        double r4531941 = r4531939 - r4531940;
        double r4531942 = r4531938 - r4531941;
        double r4531943 = r4531942 / r4531932;
        double r4531944 = 1.0;
        double r4531945 = sqrt(r4531934);
        double r4531946 = r4531944 / r4531945;
        double r4531947 = r4531931 / r4531945;
        double r4531948 = r4531946 * r4531947;
        double r4531949 = 4.0;
        double r4531950 = r4531928 * r4531928;
        double r4531951 = r4531949 / r4531950;
        double r4531952 = r4531932 / r4531928;
        double r4531953 = 8.0;
        double r4531954 = r4531928 * r4531950;
        double r4531955 = r4531953 / r4531954;
        double r4531956 = r4531952 + r4531955;
        double r4531957 = r4531951 - r4531956;
        double r4531958 = r4531948 - r4531957;
        double r4531959 = r4531958 / r4531932;
        double r4531960 = r4531930 ? r4531943 : r4531959;
        return r4531960;
}

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 < 10272132967178968.0

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-cbrt-cube0.5

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

    if 10272132967178968.0 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub49.9

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

    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. Using strategy rm

    6. Applied add-sqr-sqrt48.4

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

    7. Applied *-un-lft-identity48.4

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

    8. Applied times-frac48.4

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

    9. Taylor expanded around inf 18.4

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

    10. Simplified18.4

      \[\leadsto \frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \frac{2}{\alpha}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 10272132967178968:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))