Average Error: 3.6 → 2.3
Time: 31.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.1044946435875943 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{\frac{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}{\sqrt{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)} \cdot \frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\beta \le 1.1044946435875943 \cdot 10^{+164}:\\
\;\;\;\;\frac{\sqrt{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{\frac{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}{\sqrt{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)} \cdot \frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r2754077 = alpha;
        double r2754078 = beta;
        double r2754079 = r2754077 + r2754078;
        double r2754080 = r2754078 * r2754077;
        double r2754081 = r2754079 + r2754080;
        double r2754082 = 1.0;
        double r2754083 = r2754081 + r2754082;
        double r2754084 = 2.0;
        double r2754085 = 1.0;
        double r2754086 = r2754084 * r2754085;
        double r2754087 = r2754079 + r2754086;
        double r2754088 = r2754083 / r2754087;
        double r2754089 = r2754088 / r2754087;
        double r2754090 = r2754087 + r2754082;
        double r2754091 = r2754089 / r2754090;
        return r2754091;
}
↓
double f(double alpha, double beta) {
        double r2754092 = beta;
        double r2754093 = 1.1044946435875943e+164;
        bool r2754094 = r2754092 <= r2754093;
        double r2754095 = 1.0;
        double r2754096 = alpha;
        double r2754097 = r2754096 * r2754092;
        double r2754098 = r2754096 + r2754092;
        double r2754099 = r2754097 + r2754098;
        double r2754100 = r2754095 + r2754099;
        double r2754101 = 2.0;
        double r2754102 = r2754098 + r2754101;
        double r2754103 = r2754100 / r2754102;
        double r2754104 = r2754103 / r2754102;
        double r2754105 = sqrt(r2754104);
        double r2754106 = r2754095 + r2754102;
        double r2754107 = sqrt(r2754100);
        double r2754108 = r2754107 / r2754102;
        double r2754109 = r2754107 * r2754108;
        double r2754110 = r2754109 / r2754102;
        double r2754111 = sqrt(r2754110);
        double r2754112 = r2754106 / r2754111;
        double r2754113 = r2754105 / r2754112;
        double r2754114 = 0.0;
        double r2754115 = r2754094 ? r2754113 : r2754114;
        return r2754115;
}

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 beta < 1.1044946435875943e+164

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt1.3

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

    4. Applied associate-/l*1.3

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

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

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

    7. Applied add-sqr-sqrt1.3

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

    8. Applied times-frac1.3

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

    if 1.1044946435875943e+164 < beta

    1. Initial program 16.4

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

    2. Taylor expanded around inf 7.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.1044946435875943 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{\frac{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}{\sqrt{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)} \cdot \frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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)))