Octave 3.8, jcobi/2

Average Error: 24.1 → 1.3

Time: 20.1s

Precision: binary64

Cost: 28804

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta \cdot \frac{\beta}{\alpha}}{\alpha \cdot -0.5} + \frac{i \cdot 4 + \left(2 - \beta \cdot -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\alpha + \beta}\right)}^{-1}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

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)) -0.999998)
     (/
      (+
       (/ (* beta (/ beta alpha)) (* alpha -0.5))
       (/ (+ (* i 4.0) (- 2.0 (* beta -2.0))) alpha))
      2.0)
     (/
      (fma
       (pow (/ (+ beta (fma 2.0 i alpha)) (+ alpha beta)) -1.0)
       (/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      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)) <= -0.999998) {
		tmp = (((beta * (beta / alpha)) / (alpha * -0.5)) + (((i * 4.0) + (2.0 - (beta * -2.0))) / alpha)) / 2.0;
	} else {
		tmp = fma(pow(((beta + fma(2.0, i, alpha)) / (alpha + beta)), -1.0), ((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))), 1.0) / 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)) <= -0.999998)
		tmp = Float64(Float64(Float64(Float64(beta * Float64(beta / alpha)) / Float64(alpha * -0.5)) + Float64(Float64(Float64(i * 4.0) + Float64(2.0 - Float64(beta * -2.0))) / alpha)) / 2.0);
	else
		tmp = Float64(fma((Float64(Float64(beta + fma(2.0, i, alpha)) / Float64(alpha + beta)) ^ -1.0), Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 1.0) / 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], -0.999998], N[(N[(N[(N[(beta * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha * -0.5), $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[(N[Power[N[(N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -0.999998000000000054

   1. Initial program 62.4

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

   2. Simplified 55.3

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

      [Start] 62.4	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 62.4	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      *-commutative [=>] 62.4	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      associate-*l/ [<=] 55.3	\[ \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right)} + 1}{2} \]
      fma-def [=>] 55.3	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}, \alpha + \beta, 1\right)}}{2} \]
      *-commutative [=>] 55.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}}, \alpha + \beta, 1\right)}{2} \]
      associate-+l+ [=>] 55.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}, \alpha + \beta, 1\right)}{2} \]
      +-commutative [=>] 55.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}, \alpha + \beta, 1\right)}{2} \]
      fma-def [=>] 55.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}, \alpha + \beta, 1\right)}{2} \]
      associate-+l+ [=>] 55.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)}}, \alpha + \beta, 1\right)}{2} \]
      associate-+l+ [=>] 55.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \color{blue}{\left(\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)\right)}}, \alpha + \beta, 1\right)}{2} \]
      fma-def [=>] 55.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)\right)}, \alpha + \beta, 1\right)}{2} \]

   3. Taylor expanded in alpha around inf 13.9

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

   4. Taylor expanded in beta around inf 9.7

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

   5. Simplified 9.7

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

      [Start] 9.7	\[ \frac{-2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      associate-*r/ [=>] 9.7	\[ \frac{\color{blue}{\frac{-2 \cdot {\beta}^{2}}{{\alpha}^{2}}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 9.7	\[ \frac{\frac{-2 \cdot {\beta}^{2}}{\color{blue}{\alpha \cdot \alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      times-frac [=>] 9.7	\[ \frac{\color{blue}{\frac{-2}{\alpha} \cdot \frac{{\beta}^{2}}{\alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 9.7	\[ \frac{\frac{-2}{\alpha} \cdot \frac{\color{blue}{\beta \cdot \beta}}{\alpha} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   6. Applied egg-rr 5.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{\beta}{\alpha} \cdot \beta}{\alpha \cdot -0.5}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   if -0.999998000000000054 < (/.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 13.1

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

   2. Simplified 0.1

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

      [Start] 13.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 13.1	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 0.1	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 0.1	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 0.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 0.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 0.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 0.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 0.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 0.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 0.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

   3. Applied egg-rr 0.1

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\alpha + \beta}\right)}^{-1}}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta \cdot \frac{\beta}{\alpha}}{\alpha \cdot -0.5} + \frac{i \cdot 4 + \left(2 - \beta \cdot -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\alpha + \beta}\right)}^{-1}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	9796

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta \cdot \frac{\beta}{\alpha}}{\alpha \cdot -0.5} + \frac{i \cdot 4 + \left(2 - \beta \cdot -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1}}{2}\\ \end{array} \]

Alternative 2
Error	2.4
Cost	3140

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta \cdot \frac{\beta}{\alpha}}{\alpha \cdot -0.5} + \frac{i \cdot 4 + \left(2 - \beta \cdot -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_1}}{2}\\ \end{array} \]

Alternative 3
Error	11.4
Cost	1096

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+97}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 - \beta \cdot -2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	7.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 - \beta \cdot -2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	13.5
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.9 \cdot 10^{+98}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	17.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	24.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022356 
(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))