Average Error: 3.9 → 1.7
Time: 31.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.460679046884028130600225706603739922752 \cdot 10^{143}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \frac{\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 1.460679046884028130600225706603739922752 \cdot 10^{143}:\\
\;\;\;\;\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \frac{\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\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) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r83260 = alpha;
        double r83261 = beta;
        double r83262 = r83260 + r83261;
        double r83263 = r83261 * r83260;
        double r83264 = r83262 + r83263;
        double r83265 = 1.0;
        double r83266 = r83264 + r83265;
        double r83267 = 2.0;
        double r83268 = r83267 * r83265;
        double r83269 = r83262 + r83268;
        double r83270 = r83266 / r83269;
        double r83271 = r83270 / r83269;
        double r83272 = r83269 + r83265;
        double r83273 = r83271 / r83272;
        return r83273;
}
↓
double f(double alpha, double beta) {
        double r83274 = beta;
        double r83275 = 1.4606790468840281e+143;
        bool r83276 = r83274 <= r83275;
        double r83277 = 1.0;
        double r83278 = alpha;
        double r83279 = r83278 + r83274;
        double r83280 = 2.0;
        double r83281 = 1.0;
        double r83282 = r83280 * r83281;
        double r83283 = r83279 + r83282;
        double r83284 = sqrt(r83283);
        double r83285 = r83274 * r83278;
        double r83286 = r83279 + r83285;
        double r83287 = r83286 + r83281;
        double r83288 = r83287 / r83283;
        double r83289 = sqrt(r83288);
        double r83290 = r83284 / r83289;
        double r83291 = r83277 / r83290;
        double r83292 = r83283 + r83281;
        double r83293 = r83291 / r83292;
        double r83294 = r83291 * r83293;
        double r83295 = 2.0;
        double r83296 = r83274 / r83278;
        double r83297 = r83278 / r83274;
        double r83298 = r83296 + r83297;
        double r83299 = r83295 + r83298;
        double r83300 = r83277 / r83299;
        double r83301 = r83300 / r83292;
        double r83302 = r83276 ? r83294 : r83301;
        return r83302;
}

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 < 1.4606790468840281e+143

    1. Initial program 1.1

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

    3. Applied clear-num1.1

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

    5. Applied *-un-lft-identity1.1

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

    6. Applied add-sqr-sqrt1.9

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

    7. Applied add-sqr-sqrt1.2

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

    8. Applied times-frac1.2

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

    9. Applied add-cube-cbrt1.2

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

    10. Applied times-frac1.3

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

    11. Applied times-frac1.3

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

    12. Simplified1.3

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

    13. Simplified1.3

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

    if 1.4606790468840281e+143 < beta

    1. Initial program 16.8

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

    3. Applied clear-num16.8

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

    4. Taylor expanded around inf 3.6

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.460679046884028130600225706603739922752 \cdot 10^{143}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \frac{\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))