Octave 3.8, jcobi/4

Average Error: 52.9 → 11.7

Time: 3.4m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.4092680963047306 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}} \cdot \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}\\ \mathbf{elif}\;i \le 1.6875044460840488 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}} \cdot \frac{(\left(\alpha + \beta\right) \cdot \frac{1}{4} + \left(\frac{1}{2} \cdot i\right))_*}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}\\ \mathbf{elif}\;i \le 8.218830291734575 \cdot 10^{+182}:\\ \;\;\;\;\frac{i}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}} \cdot \frac{(\left(\alpha + \beta\right) \cdot \frac{1}{4} + \left(\frac{1}{2} \cdot i\right))_*}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r18477000 = i;
        double r18477001 = alpha;
        double r18477002 = beta;
        double r18477003 = r18477001 + r18477002;
        double r18477004 = r18477003 + r18477000;
        double r18477005 = r18477000 * r18477004;
        double r18477006 = r18477002 * r18477001;
        double r18477007 = r18477006 + r18477005;
        double r18477008 = r18477005 * r18477007;
        double r18477009 = 2.0;
        double r18477010 = r18477009 * r18477000;
        double r18477011 = r18477003 + r18477010;
        double r18477012 = r18477011 * r18477011;
        double r18477013 = r18477008 / r18477012;
        double r18477014 = 1.0;
        double r18477015 = r18477012 - r18477014;
        double r18477016 = r18477013 / r18477015;
        return r18477016;
}

↓

double f(double alpha, double beta, double i) {
        double r18477017 = i;
        double r18477018 = 4.4092680963047306e+141;
        bool r18477019 = r18477017 <= r18477018;
        double r18477020 = 1.0;
        double r18477021 = 2.0;
        double r18477022 = alpha;
        double r18477023 = beta;
        double r18477024 = r18477022 + r18477023;
        double r18477025 = fma(r18477021, r18477017, r18477024);
        double r18477026 = 1.0;
        double r18477027 = sqrt(r18477026);
        double r18477028 = r18477025 - r18477027;
        double r18477029 = r18477024 + r18477017;
        double r18477030 = r18477017 / r18477025;
        double r18477031 = r18477029 * r18477030;
        double r18477032 = r18477028 / r18477031;
        double r18477033 = r18477020 / r18477032;
        double r18477034 = r18477022 * r18477023;
        double r18477035 = fma(r18477029, r18477017, r18477034);
        double r18477036 = r18477035 / r18477025;
        double r18477037 = r18477027 + r18477025;
        double r18477038 = r18477036 / r18477037;
        double r18477039 = r18477033 * r18477038;
        double r18477040 = 1.6875044460840488e+173;
        bool r18477041 = r18477017 <= r18477040;
        double r18477042 = 0.25;
        double r18477043 = 0.5;
        double r18477044 = r18477043 * r18477017;
        double r18477045 = fma(r18477024, r18477042, r18477044);
        double r18477046 = r18477045 / r18477037;
        double r18477047 = r18477033 * r18477046;
        double r18477048 = 8.218830291734575e+182;
        bool r18477049 = r18477017 <= r18477048;
        double r18477050 = r18477017 / r18477037;
        double r18477051 = r18477050 * r18477033;
        double r18477052 = r18477049 ? r18477051 : r18477047;
        double r18477053 = r18477041 ? r18477047 : r18477052;
        double r18477054 = r18477019 ? r18477039 : r18477053;
        return r18477054;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 4.4092680963047306e+141

   1. Initial program 41.6

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

   2. Simplified 41.6

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

   4. Applied add-sqr-sqrt 41.6

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

   5. Applied difference-of-squares 41.6

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

   6. Applied times-frac 15.0

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

   7. Applied times-frac 11.0

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

   9. Applied *-un-lft-identity 11.0

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

   10. Applied associate-/l* 11.0

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

   12. Applied *-un-lft-identity 11.0

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

   13. Applied times-frac 11.0

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

   if 4.4092680963047306e+141 < i < 1.6875044460840488e+173 or 8.218830291734575e+182 < 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. Simplified 62.1

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

   4. Applied add-sqr-sqrt 62.1

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

   5. Applied difference-of-squares 62.1

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

   6. Applied times-frac 58.7

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

   7. Applied times-frac 58.6

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

   9. Applied *-un-lft-identity 58.6

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

   10. Applied associate-/l* 58.6

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

   12. Applied *-un-lft-identity 58.6

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

   13. Applied times-frac 58.6

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

   14. Taylor expanded around 0 10.5

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

   15. Simplified 10.5

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

   if 1.6875044460840488e+173 < i < 8.218830291734575e+182

   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. Simplified 62.1

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

   4. Applied add-sqr-sqrt 62.1

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

   5. Applied difference-of-squares 62.1

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

   6. Applied times-frac 62.1

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

   7. Applied times-frac 62.1

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

   9. Applied *-un-lft-identity 62.1

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

   10. Applied associate-/l* 62.1

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

   12. Applied *-un-lft-identity 62.1

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

   13. Applied times-frac 62.1

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

   14. Taylor expanded around -inf 39.0

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

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.4092680963047306 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}} \cdot \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}\\ \mathbf{elif}\;i \le 1.6875044460840488 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}} \cdot \frac{(\left(\alpha + \beta\right) \cdot \frac{1}{4} + \left(\frac{1}{2} \cdot i\right))_*}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}\\ \mathbf{elif}\;i \le 8.218830291734575 \cdot 10^{+182}:\\ \;\;\;\;\frac{i}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}} \cdot \frac{(\left(\alpha + \beta\right) \cdot \frac{1}{4} + \left(\frac{1}{2} \cdot i\right))_*}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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)))