Octave 3.8, jcobi/3

Average Error: 3.5 → 4.6

Time: 18.7s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 5.943357097504653727447872566105494970316 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{elif}\;\beta \le 3.71978879066724896437683911418213814182 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\beta \le 3.282863536225362949985859692256124964374 \cdot 10^{192}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r76089 = alpha;
        double r76090 = beta;
        double r76091 = r76089 + r76090;
        double r76092 = r76090 * r76089;
        double r76093 = r76091 + r76092;
        double r76094 = 1.0;
        double r76095 = r76093 + r76094;
        double r76096 = 2.0;
        double r76097 = r76096 * r76094;
        double r76098 = r76091 + r76097;
        double r76099 = r76095 / r76098;
        double r76100 = r76099 / r76098;
        double r76101 = r76098 + r76094;
        double r76102 = r76100 / r76101;
        return r76102;
}

↓

double f(double alpha, double beta) {
        double r76103 = beta;
        double r76104 = 5.943357097504654e+98;
        bool r76105 = r76103 <= r76104;
        double r76106 = alpha;
        double r76107 = r76106 + r76103;
        double r76108 = r76103 * r76106;
        double r76109 = r76107 + r76108;
        double r76110 = 1.0;
        double r76111 = r76109 + r76110;
        double r76112 = 2.0;
        double r76113 = r76112 * r76110;
        double r76114 = r76107 + r76113;
        double r76115 = r76111 / r76114;
        double r76116 = cbrt(r76115);
        double r76117 = sqrt(r76114);
        double r76118 = r76116 / r76117;
        double r76119 = r76114 + r76110;
        double r76120 = r76115 * r76115;
        double r76121 = cbrt(r76120);
        double r76122 = r76121 / r76117;
        double r76123 = r76119 / r76122;
        double r76124 = r76118 / r76123;
        double r76125 = 3.719788790667249e+142;
        bool r76126 = r76103 <= r76125;
        double r76127 = 0.1875;
        double r76128 = r76127 * r76106;
        double r76129 = 0.125;
        double r76130 = r76127 * r76103;
        double r76131 = r76129 + r76130;
        double r76132 = r76128 + r76131;
        double r76133 = cbrt(r76132);
        double r76134 = r76133 / r76114;
        double r76135 = r76134 / r76119;
        double r76136 = 3.282863536225363e+192;
        bool r76137 = r76103 <= r76136;
        double r76138 = cbrt(r76111);
        double r76139 = r76138 * r76138;
        double r76140 = r76114 / r76138;
        double r76141 = r76139 / r76140;
        double r76142 = r76115 * r76141;
        double r76143 = cbrt(r76142);
        double r76144 = r76143 / r76117;
        double r76145 = r76119 / r76144;
        double r76146 = r76118 / r76145;
        double r76147 = r76137 ? r76146 : r76135;
        double r76148 = r76126 ? r76135 : r76147;
        double r76149 = r76105 ? r76124 : r76148;
        return r76149;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 3 regimes

2. if beta < 5.943357097504654e+98

   1. Initial program 0.6

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

   3. Applied add-cbrt-cube 4.3

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

   4. Applied add-cbrt-cube 14.9

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

   5. Applied cbrt-undiv 14.9

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

   6. Simplified 0.6

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

   8. Applied add-sqr-sqrt 1.5

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

   9. Applied cube-mult 1.5

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

   10. Applied cbrt-prod 1.2

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

   11. Applied times-frac 0.7

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

   12. Applied associate-/l* 0.7

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

   if 5.943357097504654e+98 < beta < 3.719788790667249e+142 or 3.282863536225363e+192 < beta

   1. Initial program 14.8

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

   3. Applied add-cbrt-cube 29.3

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

   4. Applied add-cbrt-cube 62.9

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

   5. Applied cbrt-undiv 62.9

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

   6. Simplified 16.2

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

   7. Taylor expanded around 0 20.5

      \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

   if 3.719788790667249e+142 < beta < 3.282863536225363e+192

   1. Initial program 10.1

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

   3. Applied add-cbrt-cube 28.9

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

   4. Applied add-cbrt-cube 64.0

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

   5. Applied cbrt-undiv 64.0

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

   6. Simplified 12.6

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

   8. Applied add-sqr-sqrt 12.7

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

   9. Applied cube-mult 12.7

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

   10. Applied cbrt-prod 10.2

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

   11. Applied times-frac 10.2

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

   12. Applied associate-/l* 10.2

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

   14. Applied add-cube-cbrt 10.2

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

   15. Applied associate-/l* 10.2

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.943357097504653727447872566105494970316 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{elif}\;\beta \le 3.71978879066724896437683911418213814182 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\beta \le 3.282863536225362949985859692256124964374 \cdot 10^{192}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{0.1875 \cdot \alpha + \left(0.125 + 0.1875 \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(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)))