Octave 3.8, jcobi/4

Average Error: 53.9 → 11.2

Time: 2.4m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.3792583652644032 \cdot 10^{147}:\\ \;\;\;\;\frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{1} \cdot \frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{1} \cdot \frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r541150 = i;
        double r541151 = alpha;
        double r541152 = beta;
        double r541153 = r541151 + r541152;
        double r541154 = r541153 + r541150;
        double r541155 = r541150 * r541154;
        double r541156 = r541152 * r541151;
        double r541157 = r541156 + r541155;
        double r541158 = r541155 * r541157;
        double r541159 = 2.0;
        double r541160 = r541159 * r541150;
        double r541161 = r541153 + r541160;
        double r541162 = r541161 * r541161;
        double r541163 = r541158 / r541162;
        double r541164 = 1.0;
        double r541165 = r541162 - r541164;
        double r541166 = r541163 / r541165;
        return r541166;
}

↓

double f(double alpha, double beta, double i) {
        double r541167 = i;
        double r541168 = 4.379258365264403e+147;
        bool r541169 = r541167 <= r541168;
        double r541170 = sqrt(r541167);
        double r541171 = alpha;
        double r541172 = beta;
        double r541173 = r541171 + r541172;
        double r541174 = 2.0;
        double r541175 = r541174 * r541167;
        double r541176 = r541173 + r541175;
        double r541177 = 1.0;
        double r541178 = sqrt(r541177);
        double r541179 = r541176 + r541178;
        double r541180 = 1.0;
        double r541181 = r541179 / r541180;
        double r541182 = r541170 / r541181;
        double r541183 = r541182 / r541180;
        double r541184 = r541176 - r541178;
        double r541185 = r541173 + r541167;
        double r541186 = r541184 / r541185;
        double r541187 = r541170 / r541186;
        double r541188 = fma(r541167, r541174, r541173);
        double r541189 = r541167 * r541185;
        double r541190 = fma(r541172, r541171, r541189);
        double r541191 = r541190 / r541188;
        double r541192 = r541188 / r541191;
        double r541193 = r541187 / r541192;
        double r541194 = r541183 * r541193;
        double r541195 = 0.25;
        double r541196 = r541195 * r541171;
        double r541197 = 0.5;
        double r541198 = r541197 * r541167;
        double r541199 = r541195 * r541172;
        double r541200 = r541198 + r541199;
        double r541201 = r541196 + r541200;
        double r541202 = r541188 / r541201;
        double r541203 = r541187 / r541202;
        double r541204 = r541183 * r541203;
        double r541205 = r541169 ? r541194 : r541204;
        return r541205;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 4.379258365264403e+147

   1. Initial program 43.3

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

   2. Simplified 40.0

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

   4. Applied associate-/l* 32.0

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

   6. Applied *-un-lft-identity 32.0

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

   7. Applied *-un-lft-identity 32.0

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

   8. Applied times-frac 32.0

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

   9. Applied times-frac 18.5

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

   10. Applied associate-/r* 15.7

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

   11. Simplified 15.7

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

   13. Applied *-un-lft-identity 15.7

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

   14. Applied *-un-lft-identity 15.7

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

   15. Applied add-sqr-sqrt 15.7

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

   16. Applied difference-of-squares 15.7

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

   17. Applied times-frac 11.2

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

   18. Applied add-sqr-sqrt 11.6

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

   19. Applied times-frac 11.5

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

   20. Applied times-frac 11.5

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

   if 4.379258365264403e+147 < i

   1. Initial program 64.0

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

   2. Simplified 63.9

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

   4. Applied associate-/l* 63.9

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

   6. Applied *-un-lft-identity 63.9

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

   7. Applied *-un-lft-identity 63.9

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

   8. Applied times-frac 63.9

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

   9. Applied times-frac 62.4

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

   10. Applied associate-/r* 62.3

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

   11. Simplified 62.3

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

   13. Applied *-un-lft-identity 62.3

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

   14. Applied *-un-lft-identity 62.3

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

   15. Applied add-sqr-sqrt 62.3

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

   16. Applied difference-of-squares 62.3

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

   17. Applied times-frac 62.1

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

   18. Applied add-sqr-sqrt 62.2

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

   19. Applied times-frac 62.1

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

   20. Applied times-frac 62.1

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

   21. Taylor expanded around 0 10.9

       \[\leadsto \frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{1} \cdot \frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.3792583652644032 \cdot 10^{147}:\\ \;\;\;\;\frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{1} \cdot \frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{1} \cdot \frac{\frac{\sqrt{i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))