Octave 3.8, jcobi/4

Average Accuracy: 15.3% → 84.5%

Time: 35.3s

Precision: binary64

Cost: 21060

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := -1 + t_1\\ t_3 := 1 + t_1\\ \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{t_3} \cdot \frac{i}{\frac{t_2}{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.25\right)}}\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+204}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_2}{i + \alpha}}}{t_3}\\ \end{array} \]

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 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (+ -1.0 t_1))
        (t_3 (+ 1.0 t_1)))
   (if (<= beta 2.9e+99)
     (* (/ 1.0 t_3) (/ i (/ t_2 (fma 0.25 (+ beta alpha) (* i 0.25)))))
     (if (<= beta 1.75e+105)
       (/
        (/ (* i i) (/ (pow (+ beta (* i 2.0)) 2.0) (pow (+ beta i) 2.0)))
        (+ -1.0 (* t_0 t_0)))
       (if (<= beta 6.5e+204)
         (+
          (+ 0.0625 (* 0.0625 (/ (- (* 2.0 (+ beta alpha)) (+ beta alpha)) i)))
          (* (/ (+ beta alpha) i) -0.0625))
         (/ (/ i (/ t_2 (+ i alpha))) t_3))))))

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 = (beta + alpha) + (i * 2.0);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = -1.0 + t_1;
	double t_3 = 1.0 + t_1;
	double tmp;
	if (beta <= 2.9e+99) {
		tmp = (1.0 / t_3) * (i / (t_2 / fma(0.25, (beta + alpha), (i * 0.25))));
	} else if (beta <= 1.75e+105) {
		tmp = ((i * i) / (pow((beta + (i * 2.0)), 2.0) / pow((beta + i), 2.0))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 6.5e+204) {
		tmp = (0.0625 + (0.0625 * (((2.0 * (beta + alpha)) - (beta + alpha)) / i))) + (((beta + alpha) / i) * -0.0625);
	} else {
		tmp = (i / (t_2 / (i + alpha))) / t_3;
	}
	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(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(-1.0 + t_1)
	t_3 = Float64(1.0 + t_1)
	tmp = 0.0
	if (beta <= 2.9e+99)
		tmp = Float64(Float64(1.0 / t_3) * Float64(i / Float64(t_2 / fma(0.25, Float64(beta + alpha), Float64(i * 0.25)))));
	elseif (beta <= 1.75e+105)
		tmp = Float64(Float64(Float64(i * i) / Float64((Float64(beta + Float64(i * 2.0)) ^ 2.0) / (Float64(beta + i) ^ 2.0))) / Float64(-1.0 + Float64(t_0 * t_0)));
	elseif (beta <= 6.5e+204)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(2.0 * Float64(beta + alpha)) - Float64(beta + alpha)) / i))) + Float64(Float64(Float64(beta + alpha) / i) * -0.0625));
	else
		tmp = Float64(Float64(i / Float64(t_2 / Float64(i + alpha))) / t_3);
	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[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[beta, 2.9e+99], N[(N[(1.0 / t$95$3), $MachinePrecision] * N[(i / N[(t$95$2 / N[(0.25 * N[(beta + alpha), $MachinePrecision] + N[(i * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.75e+105], N[(N[(N[(i * i), $MachinePrecision] / N[(N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.5e+204], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(t$95$2 / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.9000000000000002e99

   1. Initial program 24.9%

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

   2. Taylor expanded in i around inf 45.2%

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

   3. Simplified 45.2%

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

      [Start] 45.2	\[ \frac{0.25 \cdot {i}^{2} + \left(0.25 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 45.2	\[ \frac{\color{blue}{\left(0.25 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i + 0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 45.2	\[ \frac{\left(0.25 \cdot \color{blue}{\left(2 \cdot \alpha + 2 \cdot \beta\right)} - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i + 0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [<=] 45.2	\[ \frac{\color{blue}{i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\beta + \alpha\right)\right)} + 0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 45.2	\[ \frac{\color{blue}{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\beta + \alpha\right), 0.25 \cdot {i}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 45.2	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \color{blue}{\left(2 \cdot \beta + 2 \cdot \alpha\right)} - 0.25 \cdot \left(\beta + \alpha\right), 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      distribute-lft-out-- [=>] 45.2	\[ \frac{\mathsf{fma}\left(i, \color{blue}{0.25 \cdot \left(\left(2 \cdot \beta + 2 \cdot \alpha\right) - \left(\beta + \alpha\right)\right)}, 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      distribute-lft-out [=>] 45.2	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(\color{blue}{2 \cdot \left(\beta + \alpha\right)} - \left(\beta + \alpha\right)\right), 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 45.2	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \color{blue}{{i}^{2} \cdot 0.25}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 45.2	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \color{blue}{\left(i \cdot i\right)} \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 45.3%

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

      [Start] 45.2	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-un-lft-identity [=>] 45.2	\[ \frac{\color{blue}{1 \cdot \mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 45.2	\[ \frac{1 \cdot \mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 45.2	\[ \color{blue}{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]
      +-commutative [<=] 45.2	\[ \frac{1}{\left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right) + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 45.2	\[ \frac{1}{\color{blue}{\left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 45.2	\[ \frac{1}{\left(\color{blue}{i \cdot 2} + \left(\beta + \alpha\right)\right) + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 45.2	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   5. Simplified 95.5%

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

      [Start] 45.3	\[ \frac{1}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i \cdot \left(0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      +-commutative [=>] 45.3	\[ \frac{1}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \frac{i \cdot \left(0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      associate-/l* [=>] 95.5	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}{0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25}}} \]
      +-commutative [=>] 95.5	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25}} \]
      fma-def [=>] 95.5	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\color{blue}{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.25\right)}}} \]
      *-commutative [=>] 95.5	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\mathsf{fma}\left(0.25, \beta + \alpha, \color{blue}{0.25 \cdot i}\right)}} \]

   if 2.9000000000000002e99 < beta < 1.74999999999999996e105

   1. Initial program 16.2%

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

   2. Taylor expanded in alpha around 0 14.4%

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

   3. Simplified 51.0%

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

      [Start] 14.4	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l* [=>] 51.0	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 51.0	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 51.0	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   if 1.74999999999999996e105 < beta < 6.4999999999999997e204

   1. Initial program 2.7%

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

   2. Taylor expanded in i around inf 8.0%

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

   3. Simplified 8.0%

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

      [Start] 8.0	\[ \frac{0.25 \cdot {i}^{2} + \left(0.25 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 8.0	\[ \frac{\color{blue}{\left(0.25 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i + 0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 8.0	\[ \frac{\left(0.25 \cdot \color{blue}{\left(2 \cdot \alpha + 2 \cdot \beta\right)} - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i + 0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [<=] 8.0	\[ \frac{\color{blue}{i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\beta + \alpha\right)\right)} + 0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 8.0	\[ \frac{\color{blue}{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\beta + \alpha\right), 0.25 \cdot {i}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 8.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \color{blue}{\left(2 \cdot \beta + 2 \cdot \alpha\right)} - 0.25 \cdot \left(\beta + \alpha\right), 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      distribute-lft-out-- [=>] 8.0	\[ \frac{\mathsf{fma}\left(i, \color{blue}{0.25 \cdot \left(\left(2 \cdot \beta + 2 \cdot \alpha\right) - \left(\beta + \alpha\right)\right)}, 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      distribute-lft-out [=>] 8.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(\color{blue}{2 \cdot \left(\beta + \alpha\right)} - \left(\beta + \alpha\right)\right), 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 8.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \color{blue}{{i}^{2} \cdot 0.25}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 8.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \color{blue}{\left(i \cdot i\right)} \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 52.6%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i} + 0.0625\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}} \]

   if 6.4999999999999997e204 < beta

   1. Initial program 0.0%

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

   2. Taylor expanded in beta around inf 30.0%

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

   3. Applied egg-rr 48.4%

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

      [Start] 30.0	\[ \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-un-lft-identity [=>] 30.0	\[ \frac{\color{blue}{1 \cdot \left(i \cdot \left(i + \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 30.0	\[ \frac{1 \cdot \left(i \cdot \left(i + \alpha\right)\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 48.4	\[ \color{blue}{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]
      +-commutative [=>] 48.4	\[ \frac{1}{\color{blue}{\left(2 \cdot i + \left(\alpha + \beta\right)\right)} + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 48.4	\[ \frac{1}{\left(\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 48.4	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      sub-neg [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \left(-1\right)}} \]
      +-commutative [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\left(2 \cdot i + \left(\alpha + \beta\right)\right)} + \left(-1\right)} \]
      *-commutative [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)\right) + \left(-1\right)} \]
      fma-def [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} + \left(-1\right)} \]
      metadata-eval [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + \color{blue}{-1}} \]

   4. Simplified 83.7%

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

      [Start] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*l/ [=>] 48.5	\[ \color{blue}{\frac{1 \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}} \]
      associate-/l* [=>] 83.7	\[ \frac{1 \cdot \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{i + \alpha}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \]
      +-commutative [=>] 83.7	\[ \frac{1 \cdot \frac{i}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{i + \alpha}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \]
      +-commutative [=>] 83.7	\[ \frac{1 \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \alpha}}}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.25\right)}}\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+204}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \alpha}}}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.3%
Cost	15176

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_1} \cdot \frac{i + \left(\beta + \alpha\right)}{t_1}\right)\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{-1 + t_1}{i + \alpha}}}{1 + t_1}\\ \end{array} \]

Alternative 2
Accuracy	84.3%
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+205}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{1 + t_0} \cdot \frac{i}{-1 + t_0}\\ \end{array} \]

Alternative 3
Accuracy	84.4%
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+204}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{-1 + t_0}{i + \alpha}}}{1 + t_0}\\ \end{array} \]

Alternative 4
Accuracy	84.1%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i}} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	84.1%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+205}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	83.9%
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+205}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	83.9%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+205}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + 2 \cdot \alpha}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	84.1%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+198}:\\ \;\;\;\;\left(0.0625 + \beta \cdot \frac{0.125}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 9
Accuracy	84.1%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+198}:\\ \;\;\;\;\left(0.0625 + \beta \cdot \frac{0.125}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta + i \cdot 2}\\ \end{array} \]

Alternative 10
Accuracy	83.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.26 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	83.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Accuracy	83.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 13
Accuracy	74.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+215}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 14
Accuracy	81.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 15
Accuracy	73.4%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{+248}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 16
Accuracy	70.1%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023138 
(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)))