Average Error: 54.6 → 11.7

Time: 41.9s

Precision: binary64

Cost: 4483

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

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 4.9120050759082876 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\ \mathbf{elif}\;i \leq 3.8854443850165 \cdot 10^{+112}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 1.0061247545788429 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

\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 4.9120050759082876 \cdot 10^{+54}:\\
\;\;\;\;\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} \cdot \frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\

\mathbf{elif}\;i \leq 3.8854443850165 \cdot 10^{+112}:\\
\;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\

\mathbf{elif}\;i \leq 1.0061247545788429 \cdot 10^{+137}:\\
\;\;\;\;\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} \cdot \left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\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 4.9120050759082876e+54)
   (*
    (/
     (/
      (+ (* beta alpha) (* i (+ i (+ beta alpha))))
      (+ (+ beta alpha) (* i 2.0)))
     (- (+ (+ beta alpha) (* i 2.0)) 1.0))
    (/
     (* i (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0))))
     (+ (+ (+ beta alpha) (* i 2.0)) 1.0)))
   (if (<= i 3.8854443850165e+112)
     (/
      (* 0.25 (* i i))
      (- (* (+ (+ beta alpha) (* i 2.0)) (+ (+ beta alpha) (* i 2.0))) 1.0))
     (if (<= i 1.0061247545788429e+137)
       (*
        (/
         (/
          (+ (* beta alpha) (* i (+ i (+ beta alpha))))
          (+ (+ beta alpha) (* i 2.0)))
         (- (+ (+ beta alpha) (* i 2.0)) 1.0))
        (*
         i
         (/
          (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0)))
          (+ (+ (+ beta alpha) (* i 2.0)) 1.0))))
       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 <= 4.9120050759082876e+54) {
		tmp = ((((beta * alpha) + (i * (i + (beta + alpha)))) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) - 1.0)) * ((i * ((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0)))) / (((beta + alpha) + (i * 2.0)) + 1.0));
	} else if (i <= 3.8854443850165e+112) {
		tmp = (0.25 * (i * i)) / ((((beta + alpha) + (i * 2.0)) * ((beta + alpha) + (i * 2.0))) - 1.0);
	} else if (i <= 1.0061247545788429e+137) {
		tmp = ((((beta * alpha) + (i * (i + (beta + alpha)))) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) - 1.0)) * (i * (((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) + 1.0)));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `alpha`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	38.6
Cost	44480

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}}{\sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2} \cdot \sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{\sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}}{\sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 2
Error	38.5
Cost	38080

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \left(\frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}} \cdot \frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}\right)\]

Alternative 3
Error	38.5
Cost	37568

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{\sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 4
Error	58.8
Cost	32320

\[\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\left({\left(\beta + \alpha\right)}^{3} + {\left(i \cdot 2\right)}^{3}\right) \cdot \left({\left(\beta + \alpha\right)}^{3} + {\left(i \cdot 2\right)}^{3}\right)} \cdot \left(\left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) + \left(\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - \left(\beta + \alpha\right) \cdot \left(i \cdot 2\right)\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) + \left(\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - \left(\beta + \alpha\right) \cdot \left(i \cdot 2\right)\right)\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 5
Error	54.8
Cost	27200

\[\frac{\sqrt[3]{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}} \cdot \left(\sqrt[3]{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}} \cdot \sqrt[3]{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 6
Error	38.4
Cost	25280

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{\sqrt[3]{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \left(\sqrt[3]{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \sqrt[3]{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 7
Error	55.0
Cost	25024

\[\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}} \cdot \frac{\frac{1}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}\]

Alternative 8
Error	54.8
Cost	25024

\[\frac{1}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}} \cdot \frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}\]

Alternative 9
Error	40.5
Cost	24896

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}\]

Alternative 10
Error	38.4
Cost	24768

\[\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \frac{\sqrt[3]{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \left(\sqrt[3]{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \sqrt[3]{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\]

Alternative 11
Error	61.6
Cost	24256

\[\frac{\sqrt[3]{\frac{{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}^{3} \cdot \left(\left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)\right)\right)}{{\left(\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)\right)}^{3}}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 12
Error	38.5
Cost	24256

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{\left(\sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}\right) \cdot \frac{\sqrt[3]{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 13
Error	38.5
Cost	24128

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \left(\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}} \cdot \frac{\frac{1}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}\right)\]

Alternative 14
Error	38.5
Cost	24000

Alternative 15
Error	38.5
Cost	23872

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{1}{\sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2} \cdot \sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 16
Error	38.5
Cost	23744

\[\frac{\frac{i}{\sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2} \cdot \sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2}}}{\frac{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}{\frac{i + \left(\beta + \alpha\right)}{\sqrt[3]{\left(\beta + \alpha\right) + i \cdot 2}}}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\]

Alternative 17
Error	38.5
Cost	23744

Alternative 18
Error	54.6
Cost	18496

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

Alternative 19
Error	38.2
Cost	17600

Alternative 20
Error	55.6
Cost	17600

Alternative 21
Error	38.2
Cost	17344

Alternative 22
Error	40.1
Cost	17344

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

Alternative 23
Error	38.1
Cost	17088

Alternative 24
Error	38.2
Cost	16960

Alternative 25
Error	38.2
Cost	16832

Alternative 26
Error	39.7
Cost	13760

\[\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \sqrt[3]{\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \left(\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\right)}\]

Alternative 27
Error	39.8
Cost	13248

\[\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \sqrt[3]{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right)}\]

Alternative 28
Error	62.8
Cost	8704

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

Alternative 29
Error	62.7
Cost	8576

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

Alternative 30
Error	38.1
Cost	3520

Alternative 31
Error	38.1
Cost	3520

Alternative 32
Error	38.1
Cost	3520

Alternative 33
Error	40.2
Cost	3392

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

Alternative 34
Error	40.1
Cost	3392

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

Alternative 35
Error	40.1
Cost	3392

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

Alternative 36
Error	54.6
Cost	3392

Alternative 37
Error	39.5
Cost	3136

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

Alternative 38
Error	39.6
Cost	3136

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

Alternative 39
Error	40.8
Cost	3008

Alternative 40
Error	42.7
Cost	3008

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

Alternative 41
Error	17.9
Cost	3008

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

Alternative 42
Error	54.0
Cost	2880

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

Alternative 43
Error	60.2
Cost	2752

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

Alternative 44
Error	56.3
Cost	2496

Alternative 45
Error	56.2
Cost	2496

Alternative 46
Error	46.0
Cost	2496

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

Alternative 47
Error	46.0
Cost	2496

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

Alternative 48
Error	59.3
Cost	1984

Alternative 49
Error	53.9
Cost	1984

Alternative 50
Error	54.0
Cost	1984

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

Alternative 51
Error	59.4
Cost	1984

Alternative 52
Error	18.4
Cost	1728

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

Alternative 53
Error	55.7
Cost	1472

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

Alternative 54
Error	55.5
Cost	1472

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

Alternative 55
Error	41.2
Cost	1472

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

Alternative 56
Error	59.3
Cost	576

\[\frac{i}{\frac{\beta \cdot \beta}{i + \alpha}}\]

Alternative 57
Error	59.6
Cost	576

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

Alternative 58
Error	18.3
Cost	64

\[0.0625\]

Alternative 59
Error	56.0
Cost	64

\[1\]

Alternative 60
Error	57.7
Cost	64

\[0\]

Alternative 61
Error	62.8
Cost	64

\[-1\]

Derivation

Split input into 4 regimes
if i < 4.9120050759082876e54
1. Initial program 23.2
  \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
2. Using strategy rm
3. Applied difference-of-sqr-1_binary64_237223.2
  \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}}\]
4. Applied times-frac_binary64_24089.2
  \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\]
5. Applied times-frac_binary64_24086.2
  \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}}\]
6. Simplified6.2
  \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
7. Simplified6.2
  \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \color{blue}{\frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}}\]
8. Using strategy rm
9. Applied *-un-lft-identity_binary64_24026.2
  \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
10. Applied times-frac_binary64_24086.2
  \[\leadsto \frac{\color{blue}{\frac{i}{1} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
11. Simplified6.2
  \[\leadsto \frac{\color{blue}{i} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
12. Simplified6.2
  \[\leadsto \frac{i \cdot \color{blue}{\frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
13. Simplified6.2
  \[\leadsto \color{blue}{\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}}\]
if 4.9120050759082876e54 < i < 3.8854443850165004e112
1. Initial program 50.0
  \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
2. Taylor expanded around inf 17.5
  \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
3. Simplified17.5
  \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
4. Simplified17.5
  \[\leadsto \color{blue}{\frac{0.25 \cdot \left(i \cdot i\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}\]
if 3.8854443850165004e112 < i < 1.00612475457884289e137
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_237264.0
  \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}}\]
4. Applied times-frac_binary64_240818.0
  \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\]
5. Applied times-frac_binary64_240814.7
  \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}}\]
6. Simplified14.7
  \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
7. Simplified14.7
  \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \color{blue}{\frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}}\]
8. Using strategy rm
9. Applied *-un-lft-identity_binary64_240214.7
  \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1\right)}} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
10. Applied *-un-lft-identity_binary64_240214.7
  \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1\right)} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
11. Applied times-frac_binary64_240814.6
  \[\leadsto \frac{\color{blue}{\frac{i}{1} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1\right)} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
12. Applied times-frac_binary64_240814.7
  \[\leadsto \color{blue}{\left(\frac{\frac{i}{1}}{1} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right)} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
13. Simplified14.7
  \[\leadsto \left(\color{blue}{i} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right) \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
14. Simplified14.7
  \[\leadsto \left(i \cdot \color{blue}{\frac{\frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}\right) \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
15. Simplified14.7
  \[\leadsto \color{blue}{\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right)}\]
if 1.00612475457884289e137 < 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 11.0
  \[\leadsto \color{blue}{0.0625}\]
3. Simplified11.0
  \[\leadsto \color{blue}{0.0625}\]
Recombined 4 regimes into one program.
Final simplification11.7
\[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4.9120050759082876 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\ \mathbf{elif}\;i \leq 3.8854443850165 \cdot 10^{+112}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 1.0061247545788429 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1} \cdot \left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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)))

Error

Try it out

Alternatives

Error

Derivation

`if i < 4.9120050759082876e54`

`if 4.9120050759082876e54 < i < 3.8854443850165004e112`

`if 3.8854443850165004e112 < i < 1.00612475457884289e137`

`if 1.00612475457884289e137 < i`

Reproduce