Average Error: 3.5 → 2.3
Time: 40.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.8207254528844126 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2} \cdot \left(1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\alpha + \beta\right)}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2} \cdot \frac{0.25 \cdot \left(\alpha + \beta\right) + 0.5}{4 - \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.8207254528844126 \cdot 10^{+167}:\\
\;\;\;\;\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2} \cdot \left(1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 - \left(\alpha + \beta\right)}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2} \cdot \frac{0.25 \cdot \left(\alpha + \beta\right) + 0.5}{4 - \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r3387980 = alpha;
        double r3387981 = beta;
        double r3387982 = r3387980 + r3387981;
        double r3387983 = r3387981 * r3387980;
        double r3387984 = r3387982 + r3387983;
        double r3387985 = 1.0;
        double r3387986 = r3387984 + r3387985;
        double r3387987 = 2.0;
        double r3387988 = 1.0;
        double r3387989 = r3387987 * r3387988;
        double r3387990 = r3387982 + r3387989;
        double r3387991 = r3387986 / r3387990;
        double r3387992 = r3387991 / r3387990;
        double r3387993 = r3387990 + r3387985;
        double r3387994 = r3387992 / r3387993;
        return r3387994;
}
↓
double f(double alpha, double beta) {
        double r3387995 = alpha;
        double r3387996 = 1.8207254528844126e+167;
        bool r3387997 = r3387995 <= r3387996;
        double r3387998 = 1.0;
        double r3387999 = beta;
        double r3388000 = r3387995 + r3387999;
        double r3388001 = 2.0;
        double r3388002 = r3388000 + r3388001;
        double r3388003 = r3387998 / r3388002;
        double r3388004 = 1.0;
        double r3388005 = r3387999 * r3387995;
        double r3388006 = r3388005 + r3388000;
        double r3388007 = r3388004 + r3388006;
        double r3388008 = r3388003 * r3388007;
        double r3388009 = r3388008 / r3388002;
        double r3388010 = r3388004 + r3388000;
        double r3388011 = r3388010 + r3388001;
        double r3388012 = r3388009 / r3388011;
        double r3388013 = r3388001 - r3388000;
        double r3388014 = r3388013 / r3388011;
        double r3388015 = 0.25;
        double r3388016 = r3388015 * r3388000;
        double r3388017 = 0.5;
        double r3388018 = r3388016 + r3388017;
        double r3388019 = 4.0;
        double r3388020 = r3388000 * r3388000;
        double r3388021 = r3388019 - r3388020;
        double r3388022 = r3388018 / r3388021;
        double r3388023 = r3388014 * r3388022;
        double r3388024 = r3387997 ? r3388012 : r3388023;
        return r3388024;
}

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 < 1.8207254528844126e+167

    1. Initial program 1.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified1.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]
    3. Using strategy rm

    4. Applied div-inv1.4

      \[\leadsto \frac{\frac{\color{blue}{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]
    5. Using strategy rm

    6. Applied associate-+l+1.4

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{2 + \left(\left(\beta + \alpha\right) + 1.0\right)}}\]

    if 1.8207254528844126e+167 < alpha

    1. Initial program 15.2

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified15.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]
    3. Using strategy rm

    4. Applied div-inv15.2

      \[\leadsto \frac{\frac{\color{blue}{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]
    5. Using strategy rm

    6. Applied associate-+l+15.2

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{2 + \left(\left(\beta + \alpha\right) + 1.0\right)}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity15.2

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{2 + \left(\left(\beta + \alpha\right) + \color{blue}{1 \cdot 1.0}\right)}\]

    9. Applied *-un-lft-identity15.2

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{2 + \left(\color{blue}{1 \cdot \left(\beta + \alpha\right)} + 1 \cdot 1.0\right)}\]

    10. Applied distribute-lft-out15.2

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{2 + \color{blue}{1 \cdot \left(\left(\beta + \alpha\right) + 1.0\right)}}\]

    11. Applied *-un-lft-identity15.2

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{1 \cdot 2} + 1 \cdot \left(\left(\beta + \alpha\right) + 1.0\right)}\]

    12. Applied distribute-lft-out15.2

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{1 \cdot \left(2 + \left(\left(\beta + \alpha\right) + 1.0\right)\right)}}\]

    13. Applied flip-+16.3

      \[\leadsto \frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{\color{blue}{\frac{2 \cdot 2 - \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)}{2 - \left(\beta + \alpha\right)}}}}{1 \cdot \left(2 + \left(\left(\beta + \alpha\right) + 1.0\right)\right)}\]

    14. Applied associate-/r/16.3

      \[\leadsto \frac{\color{blue}{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 \cdot 2 - \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)} \cdot \left(2 - \left(\beta + \alpha\right)\right)}}{1 \cdot \left(2 + \left(\left(\beta + \alpha\right) + 1.0\right)\right)}\]

    15. Applied times-frac16.3

      \[\leadsto \color{blue}{\frac{\frac{\left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 \cdot 2 - \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)}}{1} \cdot \frac{2 - \left(\beta + \alpha\right)}{2 + \left(\left(\beta + \alpha\right) + 1.0\right)}}\]

    16. Simplified16.3

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1.0}{\alpha + \left(2 + \beta\right)}}{4 - \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)}} \cdot \frac{2 - \left(\beta + \alpha\right)}{2 + \left(\left(\beta + \alpha\right) + 1.0\right)}\]

    17. Taylor expanded around 0 7.4

      \[\leadsto \frac{\color{blue}{0.5 + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{4 - \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)} \cdot \frac{2 - \left(\beta + \alpha\right)}{2 + \left(\left(\beta + \alpha\right) + 1.0\right)}\]

    18. Simplified7.4

      \[\leadsto \frac{\color{blue}{0.25 \cdot \left(\alpha + \beta\right) + 0.5}}{4 - \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)} \cdot \frac{2 - \left(\beta + \alpha\right)}{2 + \left(\left(\beta + \alpha\right) + 1.0\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.8207254528844126 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2} \cdot \left(1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\alpha + \beta\right)}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2} \cdot \frac{0.25 \cdot \left(\alpha + \beta\right) + 0.5}{4 - \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))