Octave 3.8, jcobi/4

Average Error: 54.0 → 11.3

Time: 2.3m

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.02606921876659773 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \mathbf{elif}\;i \le 1.5771069332023039 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}{i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r605944 = i;
        double r605945 = alpha;
        double r605946 = beta;
        double r605947 = r605945 + r605946;
        double r605948 = r605947 + r605944;
        double r605949 = r605944 * r605948;
        double r605950 = r605946 * r605945;
        double r605951 = r605950 + r605949;
        double r605952 = r605949 * r605951;
        double r605953 = 2.0;
        double r605954 = r605953 * r605944;
        double r605955 = r605947 + r605954;
        double r605956 = r605955 * r605955;
        double r605957 = r605952 / r605956;
        double r605958 = 1.0;
        double r605959 = r605956 - r605958;
        double r605960 = r605957 / r605959;
        return r605960;
}

↓

double f(double alpha, double beta, double i) {
        double r605961 = i;
        double r605962 = 1.0260692187665977e+142;
        bool r605963 = r605961 <= r605962;
        double r605964 = 2.0;
        double r605965 = alpha;
        double r605966 = beta;
        double r605967 = r605965 + r605966;
        double r605968 = fma(r605964, r605961, r605967);
        double r605969 = r605967 + r605961;
        double r605970 = r605968 / r605969;
        double r605971 = r605961 / r605970;
        double r605972 = 1.0;
        double r605973 = sqrt(r605972);
        double r605974 = r605968 + r605973;
        double r605975 = r605971 / r605974;
        double r605976 = r605964 * r605961;
        double r605977 = r605967 + r605976;
        double r605978 = r605977 - r605973;
        double r605979 = r605961 * r605969;
        double r605980 = fma(r605966, r605965, r605979);
        double r605981 = r605980 / r605968;
        double r605982 = r605978 / r605981;
        double r605983 = r605975 / r605982;
        double r605984 = 1.577106933202304e+154;
        bool r605985 = r605961 <= r605984;
        double r605986 = r605974 / r605961;
        double r605987 = r605971 / r605986;
        double r605988 = r605987 / r605978;
        double r605989 = 0.25;
        double r605990 = r605989 * r605965;
        double r605991 = 0.5;
        double r605992 = r605991 * r605961;
        double r605993 = r605989 * r605966;
        double r605994 = r605992 + r605993;
        double r605995 = r605990 + r605994;
        double r605996 = r605974 / r605995;
        double r605997 = r605971 / r605996;
        double r605998 = r605997 / r605978;
        double r605999 = r605985 ? r605988 : r605998;
        double r606000 = r605963 ? r605983 : r605999;
        return r606000;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 1.0260692187665977e+142

   1. Initial program 42.4

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

   3. Applied add-sqr-sqrt 42.4

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

   4. Applied difference-of-squares 42.4

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

   5. Applied associate-/r* 42.4

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

   6. Simplified 11.0

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

   8. Applied associate-/l* 11.0

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

   9. Simplified 11.0

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

   11. Applied associate-/l* 11.0

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

   12. Simplified 11.0

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

   14. Applied associate-/r/ 11.0

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

   15. Applied associate-/l* 11.0

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

   if 1.0260692187665977e+142 < i < 1.577106933202304e+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. Using strategy rm

   3. Applied add-sqr-sqrt 64.0

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

   4. Applied difference-of-squares 64.0

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

   5. Applied associate-/r* 64.0

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

   6. Simplified 20.9

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

   8. Applied associate-/l* 20.9

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

   9. Simplified 20.9

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

   11. Applied associate-/l* 20.9

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

   12. Simplified 20.9

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

   13. Taylor expanded around inf 38.0

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

   if 1.577106933202304e+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. Using strategy rm

   3. Applied add-sqr-sqrt 64.0

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

   4. Applied difference-of-squares 64.0

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

   5. Applied associate-/r* 64.0

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

   6. Simplified 64.0

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

   8. Applied associate-/l* 64.0

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

   9. Simplified 64.0

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

   11. Applied associate-/l* 64.0

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

   12. Simplified 64.0

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

   13. Taylor expanded around 0 9.7

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.02606921876659773 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \mathbf{elif}\;i \le 1.5771069332023039 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}{i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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)))