Average Error: 54.0 → 11.6
Time: 1.1min
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[[alpha, beta]=\mathsf{sort}([alpha, beta])\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 3.84695951150349 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{1}{\frac{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}{\sqrt{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta} \cdot \frac{\sqrt{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + i \cdot 2\right)}}}\\ \mathbf{elif}\;i \leq 3.5191541943214098 \cdot 10^{+96}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1}\\ \mathbf{elif}\;i \leq 9.287113942396901 \cdot 10^{+137}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \frac{1}{\frac{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + i \cdot 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
\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 3.84695951150349 \cdot 10^{+62}:\\
\;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{1}{\frac{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}{\sqrt{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta} \cdot \frac{\sqrt{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + i \cdot 2\right)}}}\\

\mathbf{elif}\;i \leq 3.5191541943214098 \cdot 10^{+96}:\\
\;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1}\\

\mathbf{elif}\;i \leq 9.287113942396901 \cdot 10^{+137}:\\
\;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \frac{1}{\frac{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + i \cdot 2\right)}}}\\

\mathbf{else}:\\
\;\;\;\;0.0625\\

\end{array}
(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 3.84695951150349e+62)
   (*
    (/
     (/ (* i (+ alpha (+ i beta))) (+ alpha (+ beta (* i 2.0))))
     (+ (+ alpha (+ beta (* i 2.0))) 1.0))
    (/
     1.0
     (/
      (+ alpha (- (+ beta (* i 2.0)) 1.0))
      (*
       (sqrt (+ (* i (+ alpha (+ i beta))) (* alpha beta)))
       (/
        (sqrt (+ (* i (+ alpha (+ i beta))) (* alpha beta)))
        (+ alpha (+ beta (* i 2.0))))))))
   (if (<= i 3.5191541943214098e+96)
     (/
      (* 0.25 (pow i 2.0))
      (- (* (+ (* i 2.0) (+ alpha beta)) (+ (* i 2.0) (+ alpha beta))) 1.0))
     (if (<= i 9.287113942396901e+137)
       (*
        (* (/ i (+ beta (* i 2.0))) (/ (+ i beta) (+ (* i 2.0) (+ beta 1.0))))
        (/
         1.0
         (/
          (+ alpha (- (+ beta (* i 2.0)) 1.0))
          (/
           (+ (* i (+ alpha (+ i beta))) (* alpha beta))
           (+ alpha (+ beta (* i 2.0)))))))
       0.0625))))
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 <= 3.84695951150349e+62) {
		tmp = (((i * (alpha + (i + beta))) / (alpha + (beta + (i * 2.0)))) / ((alpha + (beta + (i * 2.0))) + 1.0)) * (1.0 / ((alpha + ((beta + (i * 2.0)) - 1.0)) / (sqrt((i * (alpha + (i + beta))) + (alpha * beta)) * (sqrt((i * (alpha + (i + beta))) + (alpha * beta)) / (alpha + (beta + (i * 2.0)))))));
	} else if (i <= 3.5191541943214098e+96) {
		tmp = (0.25 * pow(i, 2.0)) / ((((i * 2.0) + (alpha + beta)) * ((i * 2.0) + (alpha + beta))) - 1.0);
	} else if (i <= 9.287113942396901e+137) {
		tmp = ((i / (beta + (i * 2.0))) * ((i + beta) / ((i * 2.0) + (beta + 1.0)))) * (1.0 / ((alpha + ((beta + (i * 2.0)) - 1.0)) / (((i * (alpha + (i + beta))) + (alpha * beta)) / (alpha + (beta + (i * 2.0))))));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if i < 3.84695951150349009e62

    1. Initial program 22.9

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

    3. Applied difference-of-sqr-1_binary64_243522.9

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

    4. Applied times-frac_binary64_247110.7

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

    5. Applied times-frac_binary64_24717.1

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

    6. Simplified7.1

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

    7. Simplified7.1

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

    9. Applied *-un-lft-identity_binary64_24657.1

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

    10. Applied *-un-lft-identity_binary64_24657.1

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

    11. Applied times-frac_binary64_24717.1

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

    12. Applied associate-/l*_binary64_24107.1

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

    13. Simplified7.1

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

    15. Applied *-un-lft-identity_binary64_24657.1

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

    16. Applied add-sqr-sqrt_binary64_24877.2

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

    17. Applied times-frac_binary64_24717.1

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

    18. Simplified7.1

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

    19. Simplified7.1

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

    if 3.84695951150349009e62 < i < 3.51915419432140976e96

    1. Initial program 49.4

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

    2. Taylor expanded around inf 18.4

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

    if 3.51915419432140976e96 < i < 9.28711394239690137e137

    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_243564.0

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

    4. Applied times-frac_binary64_247119.3

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

    5. Applied times-frac_binary64_247114.5

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

    6. Simplified14.5

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

    7. Simplified14.5

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

    9. Applied *-un-lft-identity_binary64_246514.5

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

    10. Applied *-un-lft-identity_binary64_246514.5

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

    11. Applied times-frac_binary64_247114.5

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

    12. Applied associate-/l*_binary64_241014.5

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

    13. Simplified14.5

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

    14. Taylor expanded around 0 19.3

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

    15. Simplified14.5

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

    if 9.28711394239690137e137 < 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 11.1

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.84695951150349 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{1}{\frac{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}{\sqrt{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta} \cdot \frac{\sqrt{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}}{\alpha + \left(\beta + i \cdot 2\right)}}}\\ \mathbf{elif}\;i \leq 3.5191541943214098 \cdot 10^{+96}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1}\\ \mathbf{elif}\;i \leq 9.287113942396901 \cdot 10^{+137}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \frac{1}{\frac{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + i \cdot 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(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)))