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 2.413859403597715 \cdot 10^{+114}:\\ \;\;\;\;\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{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \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 \frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \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}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{1}{4} + \frac{1}{2} \cdot i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \left(\sqrt{\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{\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)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4762088 = i;
        double r4762089 = alpha;
        double r4762090 = beta;
        double r4762091 = r4762089 + r4762090;
        double r4762092 = r4762091 + r4762088;
        double r4762093 = r4762088 * r4762092;
        double r4762094 = r4762090 * r4762089;
        double r4762095 = r4762094 + r4762093;
        double r4762096 = r4762093 * r4762095;
        double r4762097 = 2.0;
        double r4762098 = r4762097 * r4762088;
        double r4762099 = r4762091 + r4762098;
        double r4762100 = r4762099 * r4762099;
        double r4762101 = r4762096 / r4762100;
        double r4762102 = 1.0;
        double r4762103 = r4762100 - r4762102;
        double r4762104 = r4762101 / r4762103;
        return r4762104;
}

↓

double f(double alpha, double beta, double i) {
        double r4762105 = i;
        double r4762106 = 2.413859403597715e+114;
        bool r4762107 = r4762105 <= r4762106;
        double r4762108 = alpha;
        double r4762109 = beta;
        double r4762110 = r4762108 + r4762109;
        double r4762111 = 2.0;
        double r4762112 = r4762105 * r4762111;
        double r4762113 = r4762110 + r4762112;
        double r4762114 = r4762110 + r4762105;
        double r4762115 = r4762113 / r4762114;
        double r4762116 = r4762105 / r4762115;
        double r4762117 = 1.0;
        double r4762118 = sqrt(r4762117);
        double r4762119 = r4762118 + r4762113;
        double r4762120 = r4762116 / r4762119;
        double r4762121 = r4762114 * r4762105;
        double r4762122 = r4762108 * r4762109;
        double r4762123 = r4762121 + r4762122;
        double r4762124 = sqrt(r4762123);
        double r4762125 = r4762113 - r4762118;
        double r4762126 = r4762124 / r4762113;
        double r4762127 = r4762125 / r4762126;
        double r4762128 = r4762124 / r4762127;
        double r4762129 = r4762120 * r4762128;
        double r4762130 = 6.635668102175435e+131;
        bool r4762131 = r4762105 <= r4762130;
        double r4762132 = r4762105 / r4762125;
        double r4762133 = r4762120 * r4762132;
        double r4762134 = 1.1670218513000327e+144;
        bool r4762135 = r4762105 <= r4762134;
        double r4762136 = 8.296142771618488e+154;
        bool r4762137 = r4762105 <= r4762136;
        double r4762138 = 0.25;
        double r4762139 = r4762110 * r4762138;
        double r4762140 = 0.5;
        double r4762141 = r4762140 * r4762105;
        double r4762142 = r4762139 + r4762141;
        double r4762143 = r4762142 / r4762125;
        double r4762144 = sqrt(r4762120);
        double r4762145 = r4762144 * r4762144;
        double r4762146 = r4762143 * r4762145;
        double r4762147 = r4762137 ? r4762133 : r4762146;
        double r4762148 = r4762135 ? r4762129 : r4762147;
        double r4762149 = r4762131 ? r4762133 : r4762148;
        double r4762150 = r4762107 ? r4762129 : r4762149;
        return r4762150;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 2.413859403597715e+114 or 6.635668102175435e+131 < i < 1.1670218513000327e+144

   1. Initial program 39.2

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

   3. Applied add-sqr-sqrt 39.2

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

   4. Applied difference-of-squares 39.2

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

   5. Applied times-frac 15.4

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

   6. Applied times-frac 10.7

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

   8. Applied associate-/l* 10.6

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

   10. Applied *-un-lft-identity 10.6

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

   11. Applied add-sqr-sqrt 10.7

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

   12. Applied times-frac 10.6

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

   13. Applied associate-/l* 10.6

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

   if 2.413859403597715e+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* 19.6

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

   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 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

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

   12. Applied add-sqr-sqrt 9.8

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.413859403597715 \cdot 10^{+114}:\\ \;\;\;\;\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{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \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 \frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \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}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{1}{4} + \frac{1}{2} \cdot i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \left(\sqrt{\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{\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)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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)))