Octave 3.8, jcobi/2

Average Error: 24.5 → 1.4

Time: 25.6s

Precision: binary64

Cost: 28740

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}}\right)}}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.99)
     (/
      (+
       (* -2.0 (* (/ beta alpha) (/ beta alpha)))
       (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha))
      2.0)
     (/
      (exp
       (log1p
        (/
         (+ alpha beta)
         (/
          (+ alpha (+ beta (fma 2.0 i 2.0)))
          (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99) {
		tmp = ((-2.0 * ((beta / alpha) * (beta / alpha))) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0;
	} else {
		tmp = exp(log1p(((alpha + beta) / ((alpha + (beta + fma(2.0, i, 2.0))) / ((beta - alpha) / (alpha + fma(2.0, i, beta))))))) / 2.0;
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99)
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(beta / alpha) * Float64(beta / alpha))) + Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha)) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(alpha + beta) / Float64(Float64(alpha + Float64(beta + fma(2.0, i, 2.0))) / Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))))))) / 2.0);
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99], N[(N[(N[(-2.0 * N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.98999999999999999

   1. Initial program 61.7

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

   2. Simplified 55.2

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

      [Start] 61.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 13.6

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

   4. Taylor expanded in beta around inf 9.4

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

   5. Simplified 6.0

      \[\leadsto \frac{\color{blue}{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      Proof

      [Start] 9.4	\[ \frac{-2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 9.4	\[ \frac{-2 \cdot \frac{\color{blue}{\beta \cdot \beta}}{{\alpha}^{2}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 9.4	\[ \frac{-2 \cdot \frac{\beta \cdot \beta}{\color{blue}{\alpha \cdot \alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      times-frac [=>] 6.0	\[ \frac{-2 \cdot \color{blue}{\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   if -0.98999999999999999 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 13.2

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

   2. Simplified 13.2

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

      [Start] 13.2	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 13.2	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      associate-+l+ [=>] 13.2	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      associate-+l+ [=>] 13.2	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Applied egg-rr 0.0

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	16068

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]

Alternative 3
Error	1.7
Cost	15684

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(i \cdot 4 + \left(\beta + \left(2 + \alpha \cdot 2\right)\right)\right) - \alpha}\right)}}{2}\\ \end{array} \]

Alternative 4
Error	2.0
Cost	9284

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.5:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\beta}{\frac{\beta + 2 \cdot i}{\beta}}}{\beta + \mathsf{fma}\left(2, i, 2\right)}}{2}\\ \end{array} \]

Alternative 5
Error	2.2
Cost	3140

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.5:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \end{array} \]

Alternative 6
Error	13.4
Cost	1237

\[\begin{array}{l} t_0 := \frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.2 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.2 \cdot 10^{+112}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 8.2 \cdot 10^{+238} \lor \neg \left(\alpha \leq 4.6 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	13.3
Cost	1237

\[\begin{array}{l} t_0 := \frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 7 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.9 \cdot 10^{+116}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 10^{+239} \lor \neg \left(\alpha \leq 2.9 \cdot 10^{+267}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	9.6
Cost	1237

\[\begin{array}{l} t_0 := \frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.9 \cdot 10^{+115}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 9.2 \cdot 10^{+238} \lor \neg \left(\alpha \leq 7.7 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	7.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	7.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right) + \left(\beta - \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Error	15.0
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4.5 \cdot 10^{+205} \lor \neg \left(\alpha \leq 4.3 \cdot 10^{+238}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\alpha}{4}}}{2}\\ \end{array} \]

Alternative 12
Error	16.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.7 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\alpha}{4}}}{2}\\ \end{array} \]

Alternative 13
Error	18.1
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	25.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023083 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))