Average Error: 3.9 → 2.7
Time: 13.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.34518582019658975 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{-\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1 \cdot \frac{1}{\alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 2.34518582019658975 \cdot 10^{92}:\\
\;\;\;\;\frac{\frac{-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{-\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\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}\\

\end{array}
double f(double alpha, double beta) {
        double r143187 = alpha;
        double r143188 = beta;
        double r143189 = r143187 + r143188;
        double r143190 = r143188 * r143187;
        double r143191 = r143189 + r143190;
        double r143192 = 1.0;
        double r143193 = r143191 + r143192;
        double r143194 = 2.0;
        double r143195 = r143194 * r143192;
        double r143196 = r143189 + r143195;
        double r143197 = r143193 / r143196;
        double r143198 = r143197 / r143196;
        double r143199 = r143196 + r143192;
        double r143200 = r143198 / r143199;
        return r143200;
}
↓
double f(double alpha, double beta) {
        double r143201 = alpha;
        double r143202 = 2.3451858201965897e+92;
        bool r143203 = r143201 <= r143202;
        double r143204 = beta;
        double r143205 = r143201 + r143204;
        double r143206 = r143204 * r143201;
        double r143207 = r143205 + r143206;
        double r143208 = 1.0;
        double r143209 = r143207 + r143208;
        double r143210 = 2.0;
        double r143211 = r143210 * r143208;
        double r143212 = r143205 + r143211;
        double r143213 = r143209 / r143212;
        double r143214 = -r143213;
        double r143215 = -r143212;
        double r143216 = r143214 / r143215;
        double r143217 = r143212 + r143208;
        double r143218 = r143216 / r143217;
        double r143219 = 1.0;
        double r143220 = 2.0;
        double r143221 = pow(r143201, r143220);
        double r143222 = r143219 / r143221;
        double r143223 = r143210 * r143222;
        double r143224 = r143223 + r143219;
        double r143225 = r143219 / r143201;
        double r143226 = r143208 * r143225;
        double r143227 = r143224 - r143226;
        double r143228 = r143227 / r143212;
        double r143229 = r143228 / r143217;
        double r143230 = r143203 ? r143218 : r143229;
        return r143230;
}

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 < 2.3451858201965897e+92

    1. Initial program 0.5

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

    3. Applied frac-2neg0.5

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

    if 2.3451858201965897e+92 < alpha

    1. Initial program 15.1

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.34518582019658975 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{-\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1 \cdot \frac{1}{\alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))