Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 83.2%

Time: 20.3s

Precision: binary64

Cost: 16004

• Unsound rule application detected in e-graph. Results may not be sound.

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + i \cdot 2\\ \mathbf{if}\;i \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{i \cdot \frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{\left(\beta \cdot \beta + \left({t_0}^{2} + 2 \cdot \left(\beta \cdot t_0\right)\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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
 (let* ((t_0 (+ alpha (* i 2.0))))
   (if (<= i 1.6e+143)
     (/
      (/
       (* i (/ i (/ (+ beta (* i 2.0)) (+ i beta))))
       (/ (fma i 2.0 (+ beta alpha)) (+ i (+ beta alpha))))
      (+ (+ (* beta beta) (+ (pow t_0 2.0) (* 2.0 (* beta t_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 t_0 = alpha + (i * 2.0);
	double tmp;
	if (i <= 1.6e+143) {
		tmp = ((i * (i / ((beta + (i * 2.0)) / (i + beta)))) / (fma(i, 2.0, (beta + alpha)) / (i + (beta + alpha)))) / (((beta * beta) + (pow(t_0, 2.0) + (2.0 * (beta * t_0)))) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * Float64(Float64(alpha + beta) + i)) * Float64(Float64(beta * alpha) + Float64(i * Float64(Float64(alpha + beta) + i)))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i)))) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i))) - 1.0))
end

↓

function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(i * 2.0))
	tmp = 0.0
	if (i <= 1.6e+143)
		tmp = Float64(Float64(Float64(i * Float64(i / Float64(Float64(beta + Float64(i * 2.0)) / Float64(i + beta)))) / Float64(fma(i, 2.0, Float64(beta + alpha)) / Float64(i + Float64(beta + alpha)))) / Float64(Float64(Float64(beta * beta) + Float64((t_0 ^ 2.0) + Float64(2.0 * Float64(beta * t_0)))) + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 1.6e+143], N[(N[(N[(i * N[(i / N[(N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(beta * beta), $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(2.0 * N[(beta * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if i < 1.60000000000000008e143

   1. Initial program 35.3%

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

   2. Applied egg-rr 76.4%

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

      [Start] 35.3%	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      times-frac [=>] 76.4%	\[ \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 76.4%	\[ \frac{\frac{i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\color{blue}{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)} + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [<=] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \color{blue}{\alpha \cdot \beta}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-udef [<=] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Simplified 76.5%

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

      [Start] 76.4%	\[ \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l* [=>] 76.5%	\[ \frac{\color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \color{blue}{\beta + \alpha}\right)}{i + \left(\alpha + \beta\right)}} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \color{blue}{\left(\beta + \alpha\right)}}} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\color{blue}{\left(\beta + \alpha\right) + i}}} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, i + \color{blue}{\left(\beta + \alpha\right)}, \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \color{blue}{\left(\beta + \alpha\right) + i}, \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \color{blue}{\beta \cdot \alpha}\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \color{blue}{\beta + \alpha}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in beta around -inf 76.5%

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

   5. Simplified 76.5%

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

      [Start] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\left({\beta}^{2} + \left({\left(\alpha + 2 \cdot i\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + 2 \cdot i\right)\right)\right)\right) - 1} \]
      unpow2 [=>] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\left(\color{blue}{\beta \cdot \beta} + \left({\left(\alpha + 2 \cdot i\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + 2 \cdot i\right)\right)\right)\right) - 1} \]
      *-commutative [=>] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\left(\beta \cdot \beta + \left({\left(\alpha + \color{blue}{i \cdot 2}\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + 2 \cdot i\right)\right)\right)\right) - 1} \]
      *-commutative [=>] 76.5%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + \color{blue}{i \cdot 2}\right)\right)\right)\right) - 1} \]

   6. Taylor expanded in alpha around 0 76.9%

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

   7. Applied egg-rr 79.8%

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

      [Start] 76.9%	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)\right) - 1} \]
      associate-*l/ [=>] 76.9%	\[ \frac{\color{blue}{\frac{i \cdot \frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}}}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)\right) - 1} \]
      associate-/l* [=>] 79.8%	\[ \frac{\frac{i \cdot \color{blue}{\frac{i}{\frac{\beta + 2 \cdot i}{\beta + i}}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)\right) - 1} \]
      *-commutative [=>] 79.8%	\[ \frac{\frac{i \cdot \frac{i}{\frac{\beta + \color{blue}{i \cdot 2}}{\beta + i}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)\right) - 1} \]
      +-commutative [=>] 79.8%	\[ \frac{\frac{i \cdot \frac{i}{\frac{\beta + i \cdot 2}{\color{blue}{i + \beta}}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)\right) - 1} \]
      +-commutative [=>] 79.8%	\[ \frac{\frac{i \cdot \frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\color{blue}{i + \left(\beta + \alpha\right)}}}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)\right) - 1} \]

   if 1.60000000000000008e143 < i

   1. Initial program 0.1%

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

   2. Simplified 2.4%

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

      [Start] 0.1%	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 0.0%	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      associate-*l* [=>] 0.0%	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      times-frac [=>] 0.1%	\[ \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Taylor expanded in i around inf 82.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{i \cdot \frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{\left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.2%
Cost	16004

\[\begin{array}{l} t_0 := \alpha + i \cdot 2\\ \mathbf{if}\;i \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{i \cdot \frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{\left(\beta \cdot \beta + \left({t_0}^{2} + 2 \cdot \left(\beta \cdot t_0\right)\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 2
Accuracy	79.6%
Cost	16268

\[\begin{array}{l} t_0 := \alpha + i \cdot 2\\ t_1 := \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{i \cdot \left(i + \beta\right)}{\beta + i \cdot 2}\\ t_2 := \left(\beta + \alpha\right) + i \cdot 2\\ t_3 := t_2 \cdot t_2 + -1\\ \mathbf{if}\;i \leq 1.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{t_1}{t_3}\\ \mathbf{elif}\;i \leq 1.08 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{t_3}\\ \mathbf{elif}\;i \leq 1.65 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\left(\beta \cdot \beta + \left({t_0}^{2} + 2 \cdot \left(\beta \cdot t_0\right)\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 3
Accuracy	79.6%
Cost	9548

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0 + -1\\ t_2 := \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}} \cdot \frac{i \cdot \left(i + \beta\right)}{\beta + i \cdot 2}}{t_1}\\ \mathbf{if}\;i \leq 6.2 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 3.05 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{t_1}\\ \mathbf{elif}\;i \leq 1.65 \cdot 10^{+143}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 4
Accuracy	74.2%
Cost	2000

\[\begin{array}{l} t_0 := \frac{\left(i \cdot i\right) \cdot 0.25}{4 \cdot \left(i \cdot i + i \cdot \beta\right) + -1}\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := \frac{i \cdot \left(i + \alpha\right)}{t_1 \cdot t_1 + -1}\\ \mathbf{if}\;i \leq 58000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 7.1 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.3 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 5
Accuracy	73.5%
Cost	1748

\[\begin{array}{l} t_0 := \frac{\left(i \cdot i\right) \cdot 0.25}{4 \cdot \left(i \cdot i + i \cdot \beta\right) + -1}\\ \mathbf{if}\;i \leq 60000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+22}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{i + \alpha}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 6
Accuracy	78.3%
Cost	1604

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;i \leq 1.35 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{t_0 \cdot t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 7
Accuracy	74.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+220}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\frac{\beta \cdot \beta}{i}}\\ \end{array} \]

Alternative 8
Accuracy	73.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+260}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \end{array} \]

Alternative 9
Accuracy	70.8%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (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)))