Average Error: 3.5 → 2.4
Time: 51.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.00414566459986 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\frac{\left(\alpha + \beta\right) + 2}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\sqrt{\left(\alpha + \beta\right) + 2}}}}}{\left(1.0 + 2\right) + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\frac{\left(\alpha + \beta\right) + 2}{\left(\left(\alpha \cdot \sqrt{\frac{1}{2}} + \beta \cdot \sqrt{\frac{1}{2}}\right) + \sqrt{\frac{1}{2}} \cdot 1.0\right) - \left(\frac{\beta}{\sqrt{\frac{1}{2}}} \cdot 0.125 + 0.5 \cdot \left(\alpha \cdot \sqrt{\frac{1}{8}}\right)\right)}}}{\left(1.0 + 2\right) + \left(\alpha + \beta\right)}\\ \end{array}\]
\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 1.00414566459986 \cdot 10^{+210}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\frac{\left(\alpha + \beta\right) + 2}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\sqrt{\left(\alpha + \beta\right) + 2}}}}}{\left(1.0 + 2\right) + \left(\alpha + \beta\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\frac{\left(\alpha + \beta\right) + 2}{\left(\left(\alpha \cdot \sqrt{\frac{1}{2}} + \beta \cdot \sqrt{\frac{1}{2}}\right) + \sqrt{\frac{1}{2}} \cdot 1.0\right) - \left(\frac{\beta}{\sqrt{\frac{1}{2}}} \cdot 0.125 + 0.5 \cdot \left(\alpha \cdot \sqrt{\frac{1}{8}}\right)\right)}}}{\left(1.0 + 2\right) + \left(\alpha + \beta\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r4719988 = alpha;
        double r4719989 = beta;
        double r4719990 = r4719988 + r4719989;
        double r4719991 = r4719989 * r4719988;
        double r4719992 = r4719990 + r4719991;
        double r4719993 = 1.0;
        double r4719994 = r4719992 + r4719993;
        double r4719995 = 2.0;
        double r4719996 = 1.0;
        double r4719997 = r4719995 * r4719996;
        double r4719998 = r4719990 + r4719997;
        double r4719999 = r4719994 / r4719998;
        double r4720000 = r4719999 / r4719998;
        double r4720001 = r4719998 + r4719993;
        double r4720002 = r4720000 / r4720001;
        return r4720002;
}
↓
double f(double alpha, double beta) {
        double r4720003 = alpha;
        double r4720004 = 1.00414566459986e+210;
        bool r4720005 = r4720003 <= r4720004;
        double r4720006 = 1.0;
        double r4720007 = beta;
        double r4720008 = r4720003 + r4720007;
        double r4720009 = 2.0;
        double r4720010 = r4720008 + r4720009;
        double r4720011 = sqrt(r4720010);
        double r4720012 = r4720006 / r4720011;
        double r4720013 = r4720007 * r4720003;
        double r4720014 = r4720008 + r4720013;
        double r4720015 = 1.0;
        double r4720016 = r4720014 + r4720015;
        double r4720017 = r4720016 / r4720011;
        double r4720018 = r4720010 / r4720017;
        double r4720019 = r4720012 / r4720018;
        double r4720020 = r4720015 + r4720009;
        double r4720021 = r4720020 + r4720008;
        double r4720022 = r4720019 / r4720021;
        double r4720023 = 0.5;
        double r4720024 = sqrt(r4720023);
        double r4720025 = r4720003 * r4720024;
        double r4720026 = r4720007 * r4720024;
        double r4720027 = r4720025 + r4720026;
        double r4720028 = r4720024 * r4720015;
        double r4720029 = r4720027 + r4720028;
        double r4720030 = r4720007 / r4720024;
        double r4720031 = 0.125;
        double r4720032 = r4720030 * r4720031;
        double r4720033 = 0.5;
        double r4720034 = 0.125;
        double r4720035 = sqrt(r4720034);
        double r4720036 = r4720003 * r4720035;
        double r4720037 = r4720033 * r4720036;
        double r4720038 = r4720032 + r4720037;
        double r4720039 = r4720029 - r4720038;
        double r4720040 = r4720010 / r4720039;
        double r4720041 = r4720012 / r4720040;
        double r4720042 = r4720041 / r4720021;
        double r4720043 = r4720005 ? r4720022 : r4720042;
        return r4720043;
}

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 < 1.00414566459986e+210

    1. Initial program 1.9

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

    3. Applied associate-+l+1.9

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

    5. Applied add-sqr-sqrt2.4

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

    6. Applied *-un-lft-identity2.4

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

    7. Applied times-frac2.4

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

    8. Applied associate-/l*2.0

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

    if 1.00414566459986e+210 < alpha

    1. Initial program 18.1

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

    3. Applied associate-+l+18.1

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

    5. Applied add-sqr-sqrt18.1

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

    6. Applied *-un-lft-identity18.1

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

    7. Applied times-frac18.1

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

    8. Applied associate-/l*18.1

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

    9. Taylor expanded around 0 6.1

      \[\leadsto \frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\color{blue}{\left(1.0 \cdot \sqrt{\frac{1}{2}} + \left(\beta \cdot \sqrt{\frac{1}{2}} + \sqrt{\frac{1}{2}} \cdot \alpha\right)\right) - \left(0.125 \cdot \frac{\beta}{\sqrt{\frac{1}{2}}} + 0.5 \cdot \left(\sqrt{\frac{1}{8}} \cdot \alpha\right)\right)}}}}{\left(\alpha + \beta\right) + \left(2 \cdot 1 + 1.0\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.00414566459986 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\frac{\left(\alpha + \beta\right) + 2}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\sqrt{\left(\alpha + \beta\right) + 2}}}}}{\left(1.0 + 2\right) + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\frac{\left(\alpha + \beta\right) + 2}{\left(\left(\alpha \cdot \sqrt{\frac{1}{2}} + \beta \cdot \sqrt{\frac{1}{2}}\right) + \sqrt{\frac{1}{2}} \cdot 1.0\right) - \left(\frac{\beta}{\sqrt{\frac{1}{2}}} \cdot 0.125 + 0.5 \cdot \left(\alpha \cdot \sqrt{\frac{1}{8}}\right)\right)}}}{\left(1.0 + 2\right) + \left(\alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))