Average Error: 3.7 → 2.6
Time: 31.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.685848042501735519021857406529247699312 \cdot 10^{190}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 2.685848042501735519021857406529247699312 \cdot 10^{190}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}{\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 r154223 = alpha;
        double r154224 = beta;
        double r154225 = r154223 + r154224;
        double r154226 = r154224 * r154223;
        double r154227 = r154225 + r154226;
        double r154228 = 1.0;
        double r154229 = r154227 + r154228;
        double r154230 = 2.0;
        double r154231 = r154230 * r154228;
        double r154232 = r154225 + r154231;
        double r154233 = r154229 / r154232;
        double r154234 = r154233 / r154232;
        double r154235 = r154232 + r154228;
        double r154236 = r154234 / r154235;
        return r154236;
}
↓
double f(double alpha, double beta) {
        double r154237 = alpha;
        double r154238 = 2.6858480425017355e+190;
        bool r154239 = r154237 <= r154238;
        double r154240 = beta;
        double r154241 = r154237 + r154240;
        double r154242 = r154240 * r154237;
        double r154243 = r154241 + r154242;
        double r154244 = 1.0;
        double r154245 = r154243 + r154244;
        double r154246 = 2.0;
        double r154247 = r154246 * r154244;
        double r154248 = r154241 + r154247;
        double r154249 = r154245 / r154248;
        double r154250 = 3.0;
        double r154251 = pow(r154249, r154250);
        double r154252 = cbrt(r154251);
        double r154253 = r154252 / r154248;
        double r154254 = r154248 + r154244;
        double r154255 = r154253 / r154254;
        double r154256 = 0.1875;
        double r154257 = r154256 * r154237;
        double r154258 = 0.125;
        double r154259 = r154256 * r154240;
        double r154260 = r154258 + r154259;
        double r154261 = r154257 + r154260;
        double r154262 = cbrt(r154261);
        double r154263 = r154262 / r154248;
        double r154264 = r154263 / r154254;
        double r154265 = r154239 ? r154255 : r154264;
        return r154265;
}

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 < 2.6858480425017355e+190

    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 add-cbrt-cube8.8

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    4. Applied add-cbrt-cube20.0

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}}}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    5. Applied cbrt-undiv20.0

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    6. Simplified2.0

      \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    if 2.6858480425017355e+190 < alpha

    1. Initial program 18.5

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

    3. Applied add-cbrt-cube18.6

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    4. Applied add-cbrt-cube64.0

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}}}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    5. Applied cbrt-undiv64.0

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    6. Simplified18.5

      \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    7. Taylor expanded around 0 7.1

      \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.685848042501735519021857406529247699312 \cdot 10^{190}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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)))