Average Error: 16.3 → 5.9
Time: 21.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1470024490.544248104095458984375:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \left(1 - 1\right)\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1470024490.544248104095458984375:\\
\;\;\;\;e^{\log \left(\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r4149055 = beta;
        double r4149056 = alpha;
        double r4149057 = r4149055 - r4149056;
        double r4149058 = r4149056 + r4149055;
        double r4149059 = 2.0;
        double r4149060 = r4149058 + r4149059;
        double r4149061 = r4149057 / r4149060;
        double r4149062 = 1.0;
        double r4149063 = r4149061 + r4149062;
        double r4149064 = r4149063 / r4149059;
        return r4149064;
}
↓
double f(double alpha, double beta) {
        double r4149065 = alpha;
        double r4149066 = 1470024490.544248;
        bool r4149067 = r4149065 <= r4149066;
        double r4149068 = beta;
        double r4149069 = 2.0;
        double r4149070 = r4149068 + r4149065;
        double r4149071 = r4149069 + r4149070;
        double r4149072 = r4149068 / r4149071;
        double r4149073 = r4149065 / r4149071;
        double r4149074 = 1.0;
        double r4149075 = r4149073 - r4149074;
        double r4149076 = r4149072 - r4149075;
        double r4149077 = r4149076 / r4149069;
        double r4149078 = log(r4149077);
        double r4149079 = exp(r4149078);
        double r4149080 = 4.0;
        double r4149081 = r4149080 / r4149065;
        double r4149082 = r4149081 / r4149065;
        double r4149083 = 1.0;
        double r4149084 = r4149083 - r4149074;
        double r4149085 = r4149082 + r4149084;
        double r4149086 = r4149069 / r4149065;
        double r4149087 = r4149085 - r4149086;
        double r4149088 = r4149072 - r4149087;
        double r4149089 = r4149088 / r4149069;
        double r4149090 = r4149067 ? r4149079 : r4149089;
        return r4149090;
}

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 < 1470024490.544248

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-exp-log0.1

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

    7. Applied add-exp-log0.1

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

    8. Applied div-exp0.1

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

    9. Simplified0.1

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

    if 1470024490.544248 < alpha

    1. Initial program 49.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    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-log-exp48.4

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

    7. Applied add-log-exp48.4

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

    8. Applied diff-log48.5

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

    9. Taylor expanded around inf 17.9

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

    10. Simplified17.9

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1470024490.544248104095458984375:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \left(1 - 1\right)\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))