Average Error: 3.6 → 1.5
Time: 7.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 6.2833328260900076 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{1}}{1}}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\ \end{array}\]
\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 6.2833328260900076 \cdot 10^{156}:\\
\;\;\;\;\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]{1}}{1}}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r114200 = alpha;
        double r114201 = beta;
        double r114202 = r114200 + r114201;
        double r114203 = r114201 * r114200;
        double r114204 = r114202 + r114203;
        double r114205 = 1.0;
        double r114206 = r114204 + r114205;
        double r114207 = 2.0;
        double r114208 = r114207 * r114205;
        double r114209 = r114202 + r114208;
        double r114210 = r114206 / r114209;
        double r114211 = r114210 / r114209;
        double r114212 = r114209 + r114205;
        double r114213 = r114211 / r114212;
        return r114213;
}
↓
double f(double alpha, double beta) {
        double r114214 = beta;
        double r114215 = 6.2833328260900076e+156;
        bool r114216 = r114214 <= r114215;
        double r114217 = alpha;
        double r114218 = r114217 + r114214;
        double r114219 = r114214 * r114217;
        double r114220 = r114218 + r114219;
        double r114221 = 1.0;
        double r114222 = r114220 + r114221;
        double r114223 = 2.0;
        double r114224 = r114223 * r114221;
        double r114225 = r114218 + r114224;
        double r114226 = r114222 / r114225;
        double r114227 = 3.0;
        double r114228 = pow(r114226, r114227);
        double r114229 = cbrt(r114228);
        double r114230 = r114229 / r114225;
        double r114231 = r114225 + r114221;
        double r114232 = r114230 / r114231;
        double r114233 = 1.0;
        double r114234 = cbrt(r114233);
        double r114235 = r114234 / r114233;
        double r114236 = 2.0;
        double r114237 = r114214 / r114217;
        double r114238 = r114217 / r114214;
        double r114239 = r114237 + r114238;
        double r114240 = r114236 + r114239;
        double r114241 = r114240 * r114225;
        double r114242 = r114235 / r114241;
        double r114243 = r114216 ? r114232 : r114242;
        return r114243;
}

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 < 6.2833328260900076e+156

    1. Initial program 1.2

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

    3. Applied add-cbrt-cube8.3

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

    4. Applied add-cbrt-cube18.6

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

    5. Applied cbrt-undiv18.6

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

    6. Simplified1.6

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

    if 6.2833328260900076e+156 < beta

    1. Initial program 15.8

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

    3. Applied add-cbrt-cube18.2

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

    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. Simplified16.5

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

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

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

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

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

    10. Applied cbrt-prod16.5

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

    11. Applied times-frac16.5

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

    12. Applied associate-/l*16.9

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

    13. Simplified16.2

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

    14. Taylor expanded around inf 0.9

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 6.2833328260900076 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{1}}{1}}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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)))