Average Error: 3.7 → 2.3
Time: 13.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)\right) \cdot \frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;\left(0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)\right) \cdot \frac{\frac{1}{\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 r85217 = alpha;
        double r85218 = beta;
        double r85219 = r85217 + r85218;
        double r85220 = r85218 * r85217;
        double r85221 = r85219 + r85220;
        double r85222 = 1.0;
        double r85223 = r85221 + r85222;
        double r85224 = 2.0;
        double r85225 = r85224 * r85222;
        double r85226 = r85219 + r85225;
        double r85227 = r85223 / r85226;
        double r85228 = r85227 / r85226;
        double r85229 = r85226 + r85222;
        double r85230 = r85228 / r85229;
        return r85230;
}
↓
double f(double alpha, double beta) {
        double r85231 = beta;
        double r85232 = 8.48564295701033e+177;
        bool r85233 = r85231 <= r85232;
        double r85234 = alpha;
        double r85235 = r85234 + r85231;
        double r85236 = r85231 * r85234;
        double r85237 = r85235 + r85236;
        double r85238 = 1.0;
        double r85239 = r85237 + r85238;
        double r85240 = 2.0;
        double r85241 = r85240 * r85238;
        double r85242 = r85235 + r85241;
        double r85243 = r85239 / r85242;
        double r85244 = r85243 / r85242;
        double r85245 = r85242 + r85238;
        double r85246 = r85244 / r85245;
        double r85247 = 0.25;
        double r85248 = r85247 * r85234;
        double r85249 = 0.5;
        double r85250 = r85247 * r85231;
        double r85251 = r85249 + r85250;
        double r85252 = r85248 + r85251;
        double r85253 = 1.0;
        double r85254 = r85253 / r85242;
        double r85255 = r85254 / r85245;
        double r85256 = r85252 * r85255;
        double r85257 = r85233 ? r85246 : r85256;
        return r85257;
}

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 < 8.48564295701033e+177

    1. Initial program 1.6

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

    3. Applied pow11.6

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

    if 8.48564295701033e+177 < beta

    1. Initial program 16.3

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

    3. Applied pow116.3

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

    5. Applied *-un-lft-identity16.3

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

    6. Applied div-inv16.3

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

    7. Applied times-frac16.9

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

    8. Simplified16.9

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

    9. Taylor expanded around 0 6.4

      \[\leadsto \color{blue}{\left(0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)\right)} \cdot \frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)\right) \cdot \frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))