Octave 3.8, jcobi/4

Average Accuracy: 15.8% → 83.3%

Time: 30.1s

Precision: binary64

Cost: 14797.00

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\\ \mathbf{if}\;\beta \leq 7 \cdot 10^{+149}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+191} \lor \neg \left(\beta \leq 1.5 \cdot 10^{+232}\right):\\ \;\;\;\;t_1 \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.25\\ \end{array} \]

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 (fma i 2.0 (+ beta alpha)))
        (t_1 (* (/ i t_0) (/ (+ i (+ beta alpha)) t_0))))
   (if (<= beta 7e+149)
     (+ 0.0625 (/ (/ 0.015625 i) i))
     (if (or (<= beta 2.3e+191) (not (<= beta 1.5e+232)))
       (* t_1 (/ (+ i alpha) beta))
       (* t_1 0.25)))))

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 = fma(i, 2.0, (beta + alpha));
	double t_1 = (i / t_0) * ((i + (beta + alpha)) / t_0);
	double tmp;
	if (beta <= 7e+149) {
		tmp = 0.0625 + ((0.015625 / i) / i);
	} else if ((beta <= 2.3e+191) || !(beta <= 1.5e+232)) {
		tmp = t_1 * ((i + alpha) / beta);
	} else {
		tmp = t_1 * 0.25;
	}
	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 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(Float64(i / t_0) * Float64(Float64(i + Float64(beta + alpha)) / t_0))
	tmp = 0.0
	if (beta <= 7e+149)
		tmp = Float64(0.0625 + Float64(Float64(0.015625 / i) / i));
	elseif ((beta <= 2.3e+191) || !(beta <= 1.5e+232))
		tmp = Float64(t_1 * Float64(Float64(i + alpha) / beta));
	else
		tmp = Float64(t_1 * 0.25);
	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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7e+149], N[(0.0625 + N[(N[(0.015625 / i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[beta, 2.3e+191], N[Not[LessEqual[beta, 1.5e+232]], $MachinePrecision]], N[(t$95$1 * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 7.00000000000000023e149

   1. Initial program 22.8%

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

   2. Taylor expanded in i around inf 42.6%

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

   3. Simplified 42.6%

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

      [Start] 42.6	\[ \frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 42.6	\[ \frac{\color{blue}{{i}^{2} \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 42.6	\[ \frac{\color{blue}{\left(i \cdot i\right)} \cdot 0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 42.2%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot {i}^{2}} - 1} \]

   5. Simplified 42.2%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]
      Proof

      [Start] 42.2	\[ \frac{\left(i \cdot i\right) \cdot 0.25}{4 \cdot {i}^{2} - 1} \]
      *-commutative [=>] 42.2	\[ \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{{i}^{2} \cdot 4} - 1} \]
      unpow2 [=>] 42.2	\[ \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   6. Taylor expanded in i around inf 91.2%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]

   7. Simplified 91.2%

      \[\leadsto \color{blue}{0.0625 + \frac{\frac{0.015625}{i}}{i}} \]
      Proof

      [Start] 91.2	\[ 0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}} \]
      associate-*r/ [=>] 91.2	\[ 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]
      metadata-eval [=>] 91.2	\[ 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]
      unpow2 [=>] 91.2	\[ 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]
      associate-/r* [=>] 91.2	\[ 0.0625 + \color{blue}{\frac{\frac{0.015625}{i}}{i}} \]

   if 7.00000000000000023e149 < beta < 2.2999999999999999e191 or 1.50000000000000002e232 < 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. Simplified 9.5%

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

      [Start] 0.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 9.5	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 75.6%

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

   if 2.2999999999999999e191 < beta < 1.50000000000000002e232

   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. Simplified 14.8%

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

      [Start] 0.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 14.8	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 33.2%

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \color{blue}{0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+149}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+191} \lor \neg \left(\beta \leq 1.5 \cdot 10^{+232}\right):\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot 0.25\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.9%
Cost	14540.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+166}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+232}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]

Alternative 2
Accuracy	83.9%
Cost	1228.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+166}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+232}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]

Alternative 3
Accuracy	82.8%
Cost	1100.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+166}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+191}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]

Alternative 4
Accuracy	81.6%
Cost	845.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+166}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+191} \lor \neg \left(\beta \leq 1.5 \cdot 10^{+232}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 5
Accuracy	81.8%
Cost	845.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+166}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+191} \lor \neg \left(\beta \leq 1.5 \cdot 10^{+232}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 6
Accuracy	81.8%
Cost	845.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+166}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+191} \lor \neg \left(\beta \leq 1.5 \cdot 10^{+232}\right):\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 7
Accuracy	81.8%
Cost	844.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+166}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{elif}\;\beta \leq 2.2 \cdot 10^{+190}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 8
Accuracy	73.9%
Cost	196.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+264}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	9.6%
Cost	64.00

\[0 \]

Reproduce

herbie shell --seed 2023096 
(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)))