Octave 3.8, jcobi/4

Average Error: 84.31% → 14.59%

Time: 26.6s

Precision: binary64

Cost: 14541

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+82}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+96} \lor \neg \left(\beta \leq 4.9 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\frac{i}{1 + t_0}}{\frac{t_0 + -1}{i + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot 0.5\right) \cdot 0.25\\ \end{array} \]

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))))
   (if (<= beta 4.4e+82)
     (+ 0.0625 (/ 0.015625 (* i i)))
     (if (or (<= beta 5.5e+96) (not (<= beta 4.9e+171)))
       (/ (/ i (+ 1.0 t_0)) (/ (+ t_0 -1.0) (+ i alpha)))
       (* (* (/ i (+ beta (* i 2.0))) 0.5) 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 tmp;
	if (beta <= 4.4e+82) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if ((beta <= 5.5e+96) || !(beta <= 4.9e+171)) {
		tmp = (i / (1.0 + t_0)) / ((t_0 + -1.0) / (i + alpha));
	} else {
		tmp = ((i / (beta + (i * 2.0))) * 0.5) * 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))
	tmp = 0.0
	if (beta <= 4.4e+82)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif ((beta <= 5.5e+96) || !(beta <= 4.9e+171))
		tmp = Float64(Float64(i / Float64(1.0 + t_0)) / Float64(Float64(t_0 + -1.0) / Float64(i + alpha)));
	else
		tmp = Float64(Float64(Float64(i / Float64(beta + Float64(i * 2.0))) * 0.5) * 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]}, If[LessEqual[beta, 4.4e+82], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[beta, 5.5e+96], N[Not[LessEqual[beta, 4.9e+171]], $MachinePrecision]], N[(N[(i / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.4000000000000002e82

   1. Initial program 74.54

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

   2. Taylor expanded in i around inf 53.23

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

   3. Simplified 53.23

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

      [Start] 53.23	\[ \frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 53.23	\[ \frac{\color{blue}{{i}^{2} \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 53.23	\[ \frac{\color{blue}{\left(i \cdot i\right)} \cdot 0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 53.23

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

   5. Simplified 53.23

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

      [Start] 53.23	\[ \frac{\left(i \cdot i\right) \cdot 0.25}{4 \cdot {i}^{2} - 1} \]
      *-commutative [=>] 53.23	\[ \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{{i}^{2} \cdot 4} - 1} \]
      unpow2 [=>] 53.23	\[ \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   6. Taylor expanded in i around inf 3.49

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

   7. Simplified 3.49

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

      [Start] 3.49	\[ 0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}} \]
      associate-*r/ [=>] 3.49	\[ 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]
      metadata-eval [=>] 3.49	\[ 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]
      unpow2 [=>] 3.49	\[ 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   if 4.4000000000000002e82 < beta < 5.5000000000000002e96 or 4.8999999999999998e171 < beta

   1. Initial program 98.61

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

   2. Taylor expanded in beta around inf 73.44

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

   3. Applied egg-rr 53.4

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

   4. Simplified 23.78

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

      [Start] 53.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*r/ [=>] 53.4	\[ \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(i \cdot \left(i + \alpha\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}} \]
      associate-*l/ [=>] 53.36	\[ \frac{\color{blue}{\frac{1 \cdot \left(i \cdot \left(i + \alpha\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      distribute-lft-in [=>] 53.36	\[ \frac{\frac{1 \cdot \color{blue}{\left(i \cdot i + i \cdot \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      distribute-rgt-out [<=] 53.36	\[ \frac{\frac{\color{blue}{\left(i \cdot i\right) \cdot 1 + \left(i \cdot \alpha\right) \cdot 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      *-rgt-identity [=>] 53.36	\[ \frac{\frac{\color{blue}{i \cdot i} + \left(i \cdot \alpha\right) \cdot 1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      *-rgt-identity [=>] 53.36	\[ \frac{\frac{i \cdot i + \color{blue}{i \cdot \alpha}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      distribute-lft-in [<=] 53.36	\[ \frac{\frac{\color{blue}{i \cdot \left(i + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*l/ [<=] 23.76	\[ \frac{\color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(i + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-/l* [=>] 23.78	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{i + \alpha}}} \]
      +-commutative [=>] 23.78	\[ \frac{\frac{i}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{i + \alpha}} \]
      +-commutative [=>] 23.78	\[ \frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{i + \alpha}} \]

   if 5.5000000000000002e96 < beta < 4.8999999999999998e171

   1. Initial program 94.17

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

   2. Simplified 63.82

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

      [Start] 94.17	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 99.65	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 63.92	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 41.75

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

   4. Taylor expanded in i around inf 42.5

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

   5. Taylor expanded in alpha around 0 42.5

      \[\leadsto \left(\color{blue}{\frac{i}{\beta + 2 \cdot i}} \cdot 0.5\right) \cdot 0.25 \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.59

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+82}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+96} \lor \neg \left(\beta \leq 4.9 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}{i + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot 0.5\right) \cdot 0.25\\ \end{array} \]

Alternatives

Alternative 1
Error	15.32%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+86}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+98}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 5.6 \cdot 10^{+171}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot 0.5\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 2
Error	15.27%
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+86}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 1.22 \cdot 10^{+97}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 5.1 \cdot 10^{+171}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 3
Error	15.14%
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+86}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 7.3 \cdot 10^{+171}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 4
Error	17.64%
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+96} \lor \neg \left(\beta \leq 5.4 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 5
Error	17.32%
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+86}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+96} \lor \neg \left(\beta \leq 8 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 6
Error	24.58%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+210}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 7
Error	29.5%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023102 
(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)))