Octave 3.8, jcobi/4

Average Error: 54.0 → 33.8

Time: 36.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 3.473420344596471662513603003442368222614 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{elif}\;i \le 6.590410409936460439116252564895515800673 \cdot 10^{146}:\\ \;\;\;\;\frac{i + \left(\frac{1 \cdot i}{\beta \cdot \beta} - \frac{\sqrt{1} \cdot i}{\beta}\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(i \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\ \mathbf{elif}\;i \le 1.309599764436850115943908713767574659379 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}\right) \cdot \sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \left(\frac{1 \cdot i}{\beta \cdot \beta} - \frac{\sqrt{1} \cdot i}{\beta}\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(i \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r9515010 = i;
        double r9515011 = alpha;
        double r9515012 = beta;
        double r9515013 = r9515011 + r9515012;
        double r9515014 = r9515013 + r9515010;
        double r9515015 = r9515010 * r9515014;
        double r9515016 = r9515012 * r9515011;
        double r9515017 = r9515016 + r9515015;
        double r9515018 = r9515015 * r9515017;
        double r9515019 = 2.0;
        double r9515020 = r9515019 * r9515010;
        double r9515021 = r9515013 + r9515020;
        double r9515022 = r9515021 * r9515021;
        double r9515023 = r9515018 / r9515022;
        double r9515024 = 1.0;
        double r9515025 = r9515022 - r9515024;
        double r9515026 = r9515023 / r9515025;
        return r9515026;
}

↓

double f(double alpha, double beta, double i) {
        double r9515027 = i;
        double r9515028 = 3.4734203445964717e+142;
        bool r9515029 = r9515027 <= r9515028;
        double r9515030 = beta;
        double r9515031 = alpha;
        double r9515032 = r9515031 + r9515030;
        double r9515033 = r9515032 + r9515027;
        double r9515034 = r9515027 * r9515033;
        double r9515035 = fma(r9515030, r9515031, r9515034);
        double r9515036 = 2.0;
        double r9515037 = fma(r9515036, r9515027, r9515032);
        double r9515038 = 1.0;
        double r9515039 = sqrt(r9515038);
        double r9515040 = r9515037 + r9515039;
        double r9515041 = r9515035 / r9515040;
        double r9515042 = r9515037 - r9515039;
        double r9515043 = r9515034 / r9515042;
        double r9515044 = r9515043 / r9515037;
        double r9515045 = r9515041 * r9515044;
        double r9515046 = r9515045 / r9515037;
        double r9515047 = 6.59041040993646e+146;
        bool r9515048 = r9515027 <= r9515047;
        double r9515049 = r9515038 * r9515027;
        double r9515050 = r9515030 * r9515030;
        double r9515051 = r9515049 / r9515050;
        double r9515052 = r9515039 * r9515027;
        double r9515053 = r9515052 / r9515030;
        double r9515054 = r9515051 - r9515053;
        double r9515055 = r9515027 + r9515054;
        double r9515056 = r9515055 / r9515037;
        double r9515057 = r9515033 / r9515042;
        double r9515058 = r9515057 / r9515037;
        double r9515059 = r9515027 * r9515058;
        double r9515060 = r9515056 * r9515059;
        double r9515061 = 1.3095997644368501e+154;
        bool r9515062 = r9515027 <= r9515061;
        double r9515063 = sqrt(r9515044);
        double r9515064 = r9515041 * r9515063;
        double r9515065 = r9515064 * r9515063;
        double r9515066 = r9515065 / r9515037;
        double r9515067 = r9515062 ? r9515066 : r9515060;
        double r9515068 = r9515048 ? r9515060 : r9515067;
        double r9515069 = r9515029 ? r9515046 : r9515068;
        return r9515069;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 3.4734203445964717e+142

   1. Initial program 42.2

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

   2. Simplified 42.2

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

   4. Applied add-sqr-sqrt 42.2

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

   5. Applied difference-of-squares 42.2

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

   6. Applied times-frac 15.0

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

   7. Applied times-frac 10.6

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

   9. Applied associate-*l/ 10.5

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

   if 3.4734203445964717e+142 < i < 6.59041040993646e+146 or 1.3095997644368501e+154 < 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 64.0

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

   4. Applied add-sqr-sqrt 64.0

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

   5. Applied difference-of-squares 64.0

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

   6. Applied times-frac 62.8

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

   7. Applied times-frac 62.8

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

   9. Applied *-un-lft-identity 62.8

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

   10. Applied *-un-lft-identity 62.8

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

   11. Applied times-frac 62.8

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

   12. Applied times-frac 62.8

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

   13. Simplified 62.8

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

   14. Taylor expanded around inf 55.0

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

   15. Simplified 55.0

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

   if 6.59041040993646e+146 < i < 1.3095997644368501e+154

   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 64.0

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

   4. Applied add-sqr-sqrt 64.0

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

   5. Applied difference-of-squares 64.0

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

   6. Applied times-frac 20.6

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

   7. Applied times-frac 18.8

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

   9. Applied associate-*l/ 18.8

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

   11. Applied add-sqr-sqrt 18.8

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

   12. Applied associate-*r* 18.8

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

4. Final simplification 33.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.473420344596471662513603003442368222614 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{elif}\;i \le 6.590410409936460439116252564895515800673 \cdot 10^{146}:\\ \;\;\;\;\frac{i + \left(\frac{1 \cdot i}{\beta \cdot \beta} - \frac{\sqrt{1} \cdot i}{\beta}\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(i \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\ \mathbf{elif}\;i \le 1.309599764436850115943908713767574659379 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}\right) \cdot \sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \left(\frac{1 \cdot i}{\beta \cdot \beta} - \frac{\sqrt{1} \cdot i}{\beta}\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(i \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))