Octave 3.8, jcobi/4

Average Error: 52.6 → 12.5

Time: 1.2m

Precision: 64

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

Math

\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;i \le 3.744674797319727 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{elif}\;i \le 6.635668102175435 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{elif}\;i \le 1.1670218513000327 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \left(\sqrt[3]{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \left(\sqrt[3]{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \sqrt[3]{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}}\right)\right)\\ \mathbf{elif}\;i \le 8.296142771618488 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \left(\sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, i, \left(\alpha + \beta\right) \cdot \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4484927 = i;
        double r4484928 = alpha;
        double r4484929 = beta;
        double r4484930 = r4484928 + r4484929;
        double r4484931 = r4484930 + r4484927;
        double r4484932 = r4484927 * r4484931;
        double r4484933 = r4484929 * r4484928;
        double r4484934 = r4484933 + r4484932;
        double r4484935 = r4484932 * r4484934;
        double r4484936 = 2.0;
        double r4484937 = r4484936 * r4484927;
        double r4484938 = r4484930 + r4484937;
        double r4484939 = r4484938 * r4484938;
        double r4484940 = r4484935 / r4484939;
        double r4484941 = 1.0;
        double r4484942 = r4484939 - r4484941;
        double r4484943 = r4484940 / r4484942;
        return r4484943;
}

↓

double f(double alpha, double beta, double i) {
        double r4484944 = i;
        double r4484945 = 3.744674797319727e+114;
        bool r4484946 = r4484944 <= r4484945;
        double r4484947 = alpha;
        double r4484948 = beta;
        double r4484949 = r4484947 + r4484948;
        double r4484950 = 2.0;
        double r4484951 = r4484944 * r4484950;
        double r4484952 = r4484949 + r4484951;
        double r4484953 = r4484949 + r4484944;
        double r4484954 = r4484952 / r4484953;
        double r4484955 = r4484944 / r4484954;
        double r4484956 = 1.0;
        double r4484957 = sqrt(r4484956);
        double r4484958 = r4484957 + r4484952;
        double r4484959 = r4484955 / r4484958;
        double r4484960 = r4484953 * r4484944;
        double r4484961 = r4484947 * r4484948;
        double r4484962 = r4484960 + r4484961;
        double r4484963 = r4484962 / r4484952;
        double r4484964 = r4484959 * r4484963;
        double r4484965 = r4484952 - r4484957;
        double r4484966 = r4484964 / r4484965;
        double r4484967 = 6.635668102175435e+131;
        bool r4484968 = r4484944 <= r4484967;
        double r4484969 = r4484944 / r4484965;
        double r4484970 = r4484959 * r4484969;
        double r4484971 = 1.1670218513000327e+144;
        bool r4484972 = r4484944 <= r4484971;
        double r4484973 = r4484963 / r4484965;
        double r4484974 = cbrt(r4484973);
        double r4484975 = r4484974 * r4484974;
        double r4484976 = r4484974 * r4484975;
        double r4484977 = r4484959 * r4484976;
        double r4484978 = 8.296142771618488e+154;
        bool r4484979 = r4484944 <= r4484978;
        double r4484980 = cbrt(r4484959);
        double r4484981 = r4484980 * r4484980;
        double r4484982 = r4484980 * r4484981;
        double r4484983 = 0.5;
        double r4484984 = 0.25;
        double r4484985 = r4484949 * r4484984;
        double r4484986 = fma(r4484983, r4484944, r4484985);
        double r4484987 = r4484986 / r4484965;
        double r4484988 = r4484982 * r4484987;
        double r4484989 = r4484979 ? r4484970 : r4484988;
        double r4484990 = r4484972 ? r4484977 : r4484989;
        double r4484991 = r4484968 ? r4484970 : r4484990;
        double r4484992 = r4484946 ? r4484966 : r4484991;
        return r4484992;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 4 regimes

2. if i < 3.744674797319727e+114

   1. Initial program 36.9

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 36.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied difference-of-squares 36.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

   5. Applied times-frac 15.0

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}\]

   6. Applied times-frac 10.0

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   7. Using strategy rm

   8. Applied associate-/l* 10.0

      \[\leadsto \frac{\color{blue}{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
   9. Using strategy rm

   10. Applied associate-*r/ 9.9

       \[\leadsto \color{blue}{\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]

   if 3.744674797319727e+114 < i < 6.635668102175435e+131 or 1.1670218513000327e+144 < i < 8.296142771618488e+154

   1. Initial program 62.2

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 62.2

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied difference-of-squares 62.2

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

   5. Applied times-frac 23.1

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}\]

   6. Applied times-frac 19.6

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   7. Using strategy rm

   8. Applied associate-/l* 19.5

      \[\leadsto \frac{\color{blue}{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   9. Taylor expanded around -inf 36.3

      \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   if 6.635668102175435e+131 < i < 1.1670218513000327e+144

   1. Initial program 62.2

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 62.2

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied difference-of-squares 62.2

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

   5. Applied times-frac 19.5

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}\]

   6. Applied times-frac 17.1

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   7. Using strategy rm

   8. Applied associate-/l* 17.1

      \[\leadsto \frac{\color{blue}{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 17.1

       \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \sqrt[3]{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right) \cdot \sqrt[3]{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)}\]

   if 8.296142771618488e+154 < i

   1. Initial program 62.1

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied difference-of-squares 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

   5. Applied times-frac 62.1

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}\]

   6. Applied times-frac 62.1

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   7. Using strategy rm

   8. Applied associate-/l* 62.1

      \[\leadsto \frac{\color{blue}{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   9. Taylor expanded around 0 9.8

      \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   10. Simplified 9.8

       \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, i, \frac{1}{4} \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 9.8

       \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}} \cdot \sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}\right) \cdot \sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}\right)} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, i, \frac{1}{4} \cdot \left(\alpha + \beta\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.744674797319727 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{elif}\;i \le 6.635668102175435 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{elif}\;i \le 1.1670218513000327 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \left(\sqrt[3]{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \left(\sqrt[3]{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \sqrt[3]{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}}\right)\right)\\ \mathbf{elif}\;i \le 8.296142771618488 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \left(\sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \sqrt[3]{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, i, \left(\alpha + \beta\right) \cdot \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))