Average Error: 3.5 → 2.4
Time: 49.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.14701690784948810313193177185753177761 \cdot 10^{170}:\\ \;\;\;\;\sqrt{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\left(2 \cdot 1 + \left(\beta + \alpha\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\alpha}}{\alpha} + \left(1 - \frac{1}{\alpha}\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{\left(2 \cdot 1 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.14701690784948810313193177185753177761 \cdot 10^{170}:\\
\;\;\;\;\sqrt{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\left(2 \cdot 1 + \left(\beta + \alpha\right)\right) + 1}\\

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

\end{array}
double f(double alpha, double beta) {
        double r4673805 = alpha;
        double r4673806 = beta;
        double r4673807 = r4673805 + r4673806;
        double r4673808 = r4673806 * r4673805;
        double r4673809 = r4673807 + r4673808;
        double r4673810 = 1.0;
        double r4673811 = r4673809 + r4673810;
        double r4673812 = 2.0;
        double r4673813 = r4673812 * r4673810;
        double r4673814 = r4673807 + r4673813;
        double r4673815 = r4673811 / r4673814;
        double r4673816 = r4673815 / r4673814;
        double r4673817 = r4673814 + r4673810;
        double r4673818 = r4673816 / r4673817;
        return r4673818;
}
↓
double f(double alpha, double beta) {
        double r4673819 = alpha;
        double r4673820 = 2.147016907849488e+170;
        bool r4673821 = r4673819 <= r4673820;
        double r4673822 = 1.0;
        double r4673823 = beta;
        double r4673824 = r4673823 * r4673819;
        double r4673825 = r4673823 + r4673819;
        double r4673826 = r4673824 + r4673825;
        double r4673827 = r4673822 + r4673826;
        double r4673828 = 2.0;
        double r4673829 = r4673828 * r4673822;
        double r4673830 = r4673829 + r4673825;
        double r4673831 = r4673827 / r4673830;
        double r4673832 = r4673831 / r4673830;
        double r4673833 = sqrt(r4673832);
        double r4673834 = r4673830 + r4673822;
        double r4673835 = r4673833 / r4673834;
        double r4673836 = r4673833 * r4673835;
        double r4673837 = r4673828 / r4673819;
        double r4673838 = r4673837 / r4673819;
        double r4673839 = 1.0;
        double r4673840 = r4673822 / r4673819;
        double r4673841 = r4673839 - r4673840;
        double r4673842 = r4673838 + r4673841;
        double r4673843 = r4673842 / r4673830;
        double r4673844 = r4673843 / r4673834;
        double r4673845 = r4673821 ? r4673836 : r4673844;
        return r4673845;
}

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 < 2.147016907849488e+170

    1. Initial program 1.3

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

    3. Applied *-un-lft-identity1.3

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

    4. Applied add-sqr-sqrt1.4

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

    5. Applied times-frac1.4

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

    if 2.147016907849488e+170 < alpha

    1. Initial program 15.9

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    2. Taylor expanded around inf 7.9

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

    3. Simplified7.9

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.14701690784948810313193177185753177761 \cdot 10^{170}:\\ \;\;\;\;\sqrt{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\left(2 \cdot 1 + \left(\beta + \alpha\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\alpha}}{\alpha} + \left(1 - \frac{1}{\alpha}\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{\left(2 \cdot 1 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))