Octave 3.8, jcobi/4

Average Error: 54.0 → 11.3

Time: 14.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 8.5339390011673225 \cdot 10^{54}:\\ \;\;\;\;\frac{i}{\left(\beta + \left(\alpha + \mathsf{fma}\left(i, 2, \sqrt{1}\right)\right)\right) \cdot \frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{elif}\;i \le 1.3588700991283 \cdot 10^{77}:\\ \;\;\;\;\log \left(e^{\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}\right)\\ \mathbf{elif}\;i \le 8.9183161565292914 \cdot 10^{138}:\\ \;\;\;\;\frac{i}{\left(\beta + \left(\alpha + \mathsf{fma}\left(i, 2, \sqrt{1}\right)\right)\right) \cdot \frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}\\ \end{array}\]

C

double code(double alpha, double beta, double i) {
	return ((((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));
}

↓

double code(double alpha, double beta, double i) {
	double temp;
	if ((i <= 8.533939001167322e+54)) {
		temp = ((i / ((beta + (alpha + fma(i, 2.0, sqrt(1.0)))) * ((((alpha + beta) + (2.0 * i)) - sqrt(1.0)) / fma(beta, alpha, (i * ((alpha + beta) + i)))))) * ((((alpha + beta) + i) / fma(i, 2.0, (alpha + beta))) / fma(i, 2.0, (alpha + beta))));
	} else {
		double temp_1;
		if ((i <= 1.3588700991282888e+77)) {
			temp_1 = log(exp(((i / 4.0) * ((((alpha + beta) + i) / fma(i, 2.0, (alpha + beta))) / fma(i, 2.0, (alpha + beta))))));
		} else {
			double temp_2;
			if ((i <= 8.918316156529291e+138)) {
				temp_2 = ((i / ((beta + (alpha + fma(i, 2.0, sqrt(1.0)))) * ((((alpha + beta) + (2.0 * i)) - sqrt(1.0)) / fma(beta, alpha, (i * ((alpha + beta) + i)))))) * ((((alpha + beta) + i) / fma(i, 2.0, (alpha + beta))) / fma(i, 2.0, (alpha + beta))));
			} else {
				temp_2 = exp(log(((i / 4.0) * ((((alpha + beta) + i) / fma(i, 2.0, (alpha + beta))) / fma(i, 2.0, (alpha + beta))))));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 8.533939001167322e+54 or 1.3588700991282888e+77 < i < 8.918316156529291e+138

   1. Initial program 45.0

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

   2. Simplified 41.7

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

   4. Applied associate-/l* 31.9

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

   6. Applied div-inv 31.9

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

   7. Applied times-frac 18.4

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

   8. Applied times-frac 15.5

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

   9. Simplified 15.5

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

   11. Applied *-un-lft-identity 15.5

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

   12. Applied add-sqr-sqrt 15.5

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

   13. Applied difference-of-squares 15.5

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

   14. Applied times-frac 11.0

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

   15. Simplified 11.0

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

   if 8.533939001167322e+54 < i < 1.3588700991282888e+77

   1. Initial program 27.8

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

   2. Simplified 26.6

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

   4. Applied associate-/l* 23.6

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

   6. Applied div-inv 23.7

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

   7. Applied times-frac 20.7

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

   8. Applied times-frac 18.1

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

   9. Simplified 18.1

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

   10. Taylor expanded around 0 27.9

       \[\leadsto \frac{i}{\color{blue}{4}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\]
   11. Using strategy rm

   12. Applied add-log-exp 20.1

       \[\leadsto \color{blue}{\log \left(e^{\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}\right)}\]

   if 8.918316156529291e+138 < 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 div-inv 63.9

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

   7. Applied times-frac 60.2

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

   8. Applied times-frac 60.1

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

   9. Simplified 60.1

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

   10. Taylor expanded around 0 10.5

       \[\leadsto \frac{i}{\color{blue}{4}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\]
   11. Using strategy rm

   12. Applied add-exp-log 15.8

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}\]

   13. Applied add-exp-log 16.6

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   14. Applied add-exp-log 16.7

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\color{blue}{e^{\log \left(\left(\alpha + \beta\right) + i\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   15. Applied div-exp 16.7

       \[\leadsto \frac{i}{4} \cdot \frac{\color{blue}{e^{\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   16. Applied div-exp 16.7

       \[\leadsto \frac{i}{4} \cdot \color{blue}{e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   17. Applied add-exp-log 16.7

       \[\leadsto \frac{i}{\color{blue}{e^{\log 4}}} \cdot e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\]

   18. Applied add-exp-log 16.5

       \[\leadsto \frac{\color{blue}{e^{\log i}}}{e^{\log 4}} \cdot e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\]

   19. Applied div-exp 16.9

       \[\leadsto \color{blue}{e^{\log i - \log 4}} \cdot e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\]

   20. Applied prod-exp 16.9

       \[\leadsto \color{blue}{e^{\left(\log i - \log 4\right) + \left(\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right)}}\]

   21. Simplified 10.4

       \[\leadsto e^{\color{blue}{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 8.5339390011673225 \cdot 10^{54}:\\ \;\;\;\;\frac{i}{\left(\beta + \left(\alpha + \mathsf{fma}\left(i, 2, \sqrt{1}\right)\right)\right) \cdot \frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{elif}\;i \le 1.3588700991283 \cdot 10^{77}:\\ \;\;\;\;\log \left(e^{\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}\right)\\ \mathbf{elif}\;i \le 8.9183161565292914 \cdot 10^{138}:\\ \;\;\;\;\frac{i}{\left(\beta + \left(\alpha + \mathsf{fma}\left(i, 2, \sqrt{1}\right)\right)\right) \cdot \frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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)))