Octave 3.8, jcobi/4

Average Error: 54.0 → 11.4

Time: 13.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.8129575196524918 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} - \frac{1}{\left(\alpha + \beta\right) + i}}}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\\ \mathbf{elif}\;i \le 5.44489365468635089 \cdot 10^{130}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} - \frac{1}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} - \frac{1}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, 0.25 \cdot \beta\right)\right)}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r115608 = i;
        double r115609 = alpha;
        double r115610 = beta;
        double r115611 = r115609 + r115610;
        double r115612 = r115611 + r115608;
        double r115613 = r115608 * r115612;
        double r115614 = r115610 * r115609;
        double r115615 = r115614 + r115613;
        double r115616 = r115613 * r115615;
        double r115617 = 2.0;
        double r115618 = r115617 * r115608;
        double r115619 = r115611 + r115618;
        double r115620 = r115619 * r115619;
        double r115621 = r115616 / r115620;
        double r115622 = 1.0;
        double r115623 = r115620 - r115622;
        double r115624 = r115621 / r115623;
        return r115624;
}

↓

double f(double alpha, double beta, double i) {
        double r115625 = i;
        double r115626 = 4.812957519652492e+123;
        bool r115627 = r115625 <= r115626;
        double r115628 = 2.0;
        double r115629 = alpha;
        double r115630 = beta;
        double r115631 = r115629 + r115630;
        double r115632 = fma(r115625, r115628, r115631);
        double r115633 = r115631 + r115625;
        double r115634 = r115633 / r115632;
        double r115635 = r115632 / r115634;
        double r115636 = 1.0;
        double r115637 = r115636 / r115633;
        double r115638 = r115635 - r115637;
        double r115639 = r115625 / r115638;
        double r115640 = r115625 * r115633;
        double r115641 = fma(r115630, r115629, r115640);
        double r115642 = r115632 / r115641;
        double r115643 = 1.0;
        double r115644 = r115643 / r115632;
        double r115645 = r115642 / r115644;
        double r115646 = r115639 / r115645;
        double r115647 = 5.444893654686351e+130;
        bool r115648 = r115625 <= r115647;
        double r115649 = r115632 / r115625;
        double r115650 = r115639 / r115649;
        double r115651 = 0.25;
        double r115652 = 0.5;
        double r115653 = r115651 * r115630;
        double r115654 = fma(r115652, r115625, r115653);
        double r115655 = fma(r115651, r115629, r115654);
        double r115656 = r115632 / r115655;
        double r115657 = r115639 / r115656;
        double r115658 = r115648 ? r115650 : r115657;
        double r115659 = r115627 ? r115646 : r115658;
        return r115659;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 4.812957519652492e+123

   1. Initial program 39.3

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

   2. Simplified 36.6

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

   4. Applied associate-/l* 26.9

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

   6. Applied *-un-lft-identity 26.9

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

   7. Applied *-un-lft-identity 26.9

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

   8. Applied times-frac 26.9

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

   9. Applied times-frac 18.1

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

   10. Applied associate-/r* 14.7

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

   11. Simplified 14.6

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

   13. Applied div-sub 14.6

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

   14. Simplified 10.1

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

   16. Applied div-inv 10.3

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

   17. Applied associate-/r* 10.3

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

   if 4.812957519652492e+123 < i < 5.444893654686351e+130

   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 62.5

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

   4. Applied associate-/l* 62.5

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

   6. Applied *-un-lft-identity 62.5

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

   7. Applied *-un-lft-identity 62.5

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

   8. Applied times-frac 62.5

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

   9. Applied times-frac 24.5

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

   10. Applied associate-/r* 23.4

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

   11. Simplified 23.3

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

   13. Applied div-sub 23.3

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

   14. Simplified 18.6

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

   15. Taylor expanded around inf 33.9

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

   if 5.444893654686351e+130 < 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 63.8

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

   4. Applied associate-/l* 63.8

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

   6. Applied *-un-lft-identity 63.8

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

   7. Applied *-un-lft-identity 63.8

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

   8. Applied times-frac 63.8

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

   9. Applied times-frac 58.3

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

   10. Applied associate-/r* 58.1

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

   11. Simplified 58.1

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

   13. Applied div-sub 58.1

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

   14. Simplified 57.9

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

   15. Taylor expanded around 0 11.5

       \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} - \frac{1}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}}\]

   16. Simplified 11.5

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

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.8129575196524918 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} - \frac{1}{\left(\alpha + \beta\right) + i}}}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\\ \mathbf{elif}\;i \le 5.44489365468635089 \cdot 10^{130}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} - \frac{1}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} - \frac{1}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, 0.25 \cdot \beta\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))