Octave 3.8, jcobi/4

Average Error: 53.8 → 11.0

Time: 25.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 2.5206724026500415 \cdot 10^{+85}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 6.09812200291671 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 2.061938149421035 \cdot 10^{+144}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* 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)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.5206724026500415e+85)
   (*
    (/
     (* i (/ (+ i (+ alpha beta)) (+ (+ alpha beta) (* i 2.0))))
     (+ (+ (+ alpha beta) (* i 2.0)) 1.0))
    (/
     (/
      (+ (* alpha beta) (* i (+ i (+ alpha beta))))
      (+ (+ alpha beta) (* i 2.0)))
     (- (+ (+ alpha beta) (* i 2.0)) 1.0)))
   (if (<= i 6.09812200291671e+143)
     (/
      (* (* i i) 0.25)
      (- (* (+ (+ alpha beta) (* i 2.0)) (+ (+ alpha beta) (* i 2.0))) 1.0))
     (if (<= i 2.061938149421035e+144)
       (exp
        (log
         (*
          (/
           (/
            (+ (* alpha beta) (* i (+ i (+ alpha beta))))
            (+ (+ alpha beta) (* i 2.0)))
           (- (+ (+ alpha beta) (* i 2.0)) 1.0))
          (/
           (/ (* i (+ i (+ alpha beta))) (+ (+ alpha beta) (* i 2.0)))
           (+ (+ (+ alpha beta) (* i 2.0)) 1.0)))))
       0.0625))))

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 tmp;
	if (i <= 2.5206724026500415e+85) {
		tmp = ((i * ((i + (alpha + beta)) / ((alpha + beta) + (i * 2.0)))) / (((alpha + beta) + (i * 2.0)) + 1.0)) * ((((alpha * beta) + (i * (i + (alpha + beta)))) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) - 1.0));
	} else if (i <= 6.09812200291671e+143) {
		tmp = ((i * i) * 0.25) / ((((alpha + beta) + (i * 2.0)) * ((alpha + beta) + (i * 2.0))) - 1.0);
	} else if (i <= 2.061938149421035e+144) {
		tmp = exp(log(((((alpha * beta) + (i * (i + (alpha + beta)))) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) - 1.0)) * (((i * (i + (alpha + beta))) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) + 1.0))));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 4 regimes

2. if i < 2.5206724026500415e85

   1. Initial program 28.5

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

   3. Applied difference-of-sqr-1_binary64_1753 28.5

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

   4. Applied times-frac_binary64_1789 12.4

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

   5. Applied times-frac_binary64_1789 7.6

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

   6. Simplified 7.6

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

   7. Simplified 7.6

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

   9. Applied *-un-lft-identity_binary64_1783 7.6

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

   10. Applied times-frac_binary64_1789 7.5

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

   11. Simplified 7.5

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

   if 2.5206724026500415e85 < i < 6.0981220029167096e143

   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. Taylor expanded around inf 18.5

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

   3. Simplified 18.5

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

   if 6.0981220029167096e143 < i < 2.0619381494210352e144

   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 difference-of-sqr-1_binary64_1753 64.0

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

   4. Applied times-frac_binary64_1789 21.3

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

   5. Applied times-frac_binary64_1789 21.3

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

   6. Simplified 21.3

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

   7. Simplified 21.3

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

   9. Applied add-exp-log_binary64_1821 25.5

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

   10. Applied add-exp-log_binary64_1821 26.2

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

   11. Applied add-exp-log_binary64_1821 24.7

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

   12. Applied div-exp_binary64_1834 24.7

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

   13. Applied div-exp_binary64_1834 24.8

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

   14. Applied add-exp-log_binary64_1821 25.3

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

   15. Applied add-exp-log_binary64_1821 26.1

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

   16. Applied add-exp-log_binary64_1821 25.1

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

   17. Applied add-exp-log_binary64_1821 25.4

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

   18. Applied prod-exp_binary64_1832 25.4

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

   19. Applied div-exp_binary64_1834 25.4

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

   20. Applied div-exp_binary64_1834 25.4

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

   21. Applied prod-exp_binary64_1832 25.4

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

   22. Simplified 21.3

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

   if 2.0619381494210352e144 < 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. Taylor expanded around inf 10.3

      \[\leadsto \color{blue}{0.0625}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.5206724026500415 \cdot 10^{+85}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 6.09812200291671 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 2.061938149421035 \cdot 10^{+144}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2021009 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))