Average Error: 53.8 → 17.0
Time: 19.4s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq -9.765822633999939 \cdot 10^{-119}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq -2.139573966778813 \cdot 10^{-173}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;\beta \leq 2.0833052166247658 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 3.173964194173191 \cdot 10^{+157}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;\beta \leq 1.921741030358534 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \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}\;\beta \leq -9.765822633999939 \cdot 10^{-119}:\\
\;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\

\mathbf{elif}\;\beta \leq -2.139573966778813 \cdot 10^{-173}:\\
\;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\

\mathbf{elif}\;\beta \leq 2.0833052166247658 \cdot 10^{+88}:\\
\;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\

\mathbf{elif}\;\beta \leq 3.173964194173191 \cdot 10^{+157}:\\
\;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\

\mathbf{elif}\;\beta \leq 1.921741030358534 \cdot 10^{+212}:\\
\;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\

\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 (<= beta -9.765822633999939e-119)
   (/
    (*
     (* i (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (+ i i))))
     (/
      (+ (* i 0.5) (* (+ beta alpha) 0.25))
      (- (+ (+ beta alpha) (+ i i)) 1.0)))
    (+ 1.0 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta -2.139573966778813e-173)
     (*
      (/
       (* i (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0))))
       (+ 1.0 (+ (+ beta alpha) (* i 2.0))))
      (/ i (- (+ (+ beta alpha) (* i 2.0)) 1.0)))
     (if (<= beta 2.0833052166247658e+88)
       (/
        (*
         (* i (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (+ i i))))
         (/
          (+ (* i 0.5) (* (+ beta alpha) 0.25))
          (- (+ (+ beta alpha) (+ i i)) 1.0)))
        (+ 1.0 (+ (+ beta alpha) (* i 2.0))))
       (if (<= beta 3.173964194173191e+157)
         (*
          (/
           (* i (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0))))
           (+ 1.0 (+ (+ beta alpha) (* i 2.0))))
          (/
           (/
            (+ (* beta alpha) (* i (+ i (+ beta alpha))))
            (+ (+ beta alpha) (* i 2.0)))
           (- (+ (+ beta alpha) (* i 2.0)) 1.0)))
         (if (<= beta 1.921741030358534e+212)
           (/
            (*
             (* i (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (+ i i))))
             (/
              (+ (* i 0.5) (* (+ beta alpha) 0.25))
              (- (+ (+ beta alpha) (+ i i)) 1.0)))
            (+ 1.0 (+ (+ beta alpha) (* i 2.0))))
           (*
            (/
             (* i (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0))))
             (+ 1.0 (+ (+ beta alpha) (* i 2.0))))
            (/ i (- (+ (+ beta alpha) (* i 2.0)) 1.0)))))))))
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 (beta <= -9.765822633999939e-119) {
		tmp = ((i * ((i + (beta + alpha)) / ((beta + alpha) + (i + i)))) * (((i * 0.5) + ((beta + alpha) * 0.25)) / (((beta + alpha) + (i + i)) - 1.0))) / (1.0 + ((beta + alpha) + (i * 2.0)));
	} else if (beta <= -2.139573966778813e-173) {
		tmp = ((i * ((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0)))) / (1.0 + ((beta + alpha) + (i * 2.0)))) * (i / (((beta + alpha) + (i * 2.0)) - 1.0));
	} else if (beta <= 2.0833052166247658e+88) {
		tmp = ((i * ((i + (beta + alpha)) / ((beta + alpha) + (i + i)))) * (((i * 0.5) + ((beta + alpha) * 0.25)) / (((beta + alpha) + (i + i)) - 1.0))) / (1.0 + ((beta + alpha) + (i * 2.0)));
	} else if (beta <= 3.173964194173191e+157) {
		tmp = ((i * ((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0)))) / (1.0 + ((beta + alpha) + (i * 2.0)))) * ((((beta * alpha) + (i * (i + (beta + alpha)))) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) - 1.0));
	} else if (beta <= 1.921741030358534e+212) {
		tmp = ((i * ((i + (beta + alpha)) / ((beta + alpha) + (i + i)))) * (((i * 0.5) + ((beta + alpha) * 0.25)) / (((beta + alpha) + (i + i)) - 1.0))) / (1.0 + ((beta + alpha) + (i * 2.0)));
	} else {
		tmp = ((i * ((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0)))) / (1.0 + ((beta + alpha) + (i * 2.0)))) * (i / (((beta + alpha) + (i * 2.0)) - 1.0));
	}
	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 3 regimes

  2. if beta < -9.76582263399993908e-119 or -2.139573966778813e-173 < beta < 2.0833052166247658e88 or 3.17396419417319119e157 < beta < 1.92174103035853415e212

    1. Initial program 52.1

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

    3. Applied difference-of-sqr-1_binary6452.1

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

    4. Applied times-frac_binary6437.3

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

    5. Applied times-frac_binary6435.0

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

    6. Simplified35.0

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

    7. Simplified35.0

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

    9. Applied *-un-lft-identity_binary6435.0

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

    10. Applied times-frac_binary6435.0

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

    11. Simplified35.0

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

    12. Taylor expanded around 0 13.2

      \[\leadsto \frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]

    13. Simplified13.2

      \[\leadsto \frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\color{blue}{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
    14. Using strategy rm

    15. Applied associate-*l/_binary6413.2

      \[\leadsto \color{blue}{\frac{\left(i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}\right) \cdot \frac{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}\]

    16. Simplified13.2

      \[\leadsto \frac{\color{blue}{\left(i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + 0.25 \cdot \left(\alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + \left(i + i\right)\right) - 1}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\]

    if -9.76582263399993908e-119 < beta < -2.139573966778813e-173 or 1.92174103035853415e212 < beta

    1. Initial program 59.3

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

    3. Applied difference-of-sqr-1_binary6459.3

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

    4. Applied times-frac_binary6449.3

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

    5. Applied times-frac_binary6448.2

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

    6. Simplified48.2

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

    7. Simplified48.2

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

    9. Applied *-un-lft-identity_binary6448.2

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

    10. Applied times-frac_binary6448.2

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

    11. Simplified48.2

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

    12. Taylor expanded around inf 26.6

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

    if 2.0833052166247658e88 < beta < 3.17396419417319119e157

    1. Initial program 59.9

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

    3. Applied difference-of-sqr-1_binary6459.9

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

    4. Applied times-frac_binary6440.0

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

    5. Applied times-frac_binary6437.5

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

    6. Simplified37.5

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

    7. Simplified37.5

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

    9. Applied *-un-lft-identity_binary6437.5

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

    10. Applied times-frac_binary6437.4

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

    11. Simplified37.4

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -9.765822633999939 \cdot 10^{-119}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq -2.139573966778813 \cdot 10^{-173}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;\beta \leq 2.0833052166247658 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 3.173964194173191 \cdot 10^{+157}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;\beta \leq 1.921741030358534 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + \left(i + i\right)}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right) - 1}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(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)))