Octave 3.8, jcobi/2

Average Accuracy: 63.0% → 98.1%

Time: 26.4s

Precision: binary64

Cost: 35268

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} + \frac{\beta - -2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 2\right)}, \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

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)) -0.999999)
     (/ (+ (/ beta alpha) (+ (/ (* i 4.0) alpha) (/ (- beta -2.0) alpha))) 2.0)
     (/
      (pow
       (pow
        (fma
         (/ (- beta alpha) (fma 2.0 i (+ (+ alpha beta) 2.0)))
         (/ (+ alpha beta) (+ alpha (fma 2.0 i beta)))
         1.0)
        3.0)
       0.3333333333333333)
      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)) <= -0.999999) {
		tmp = ((beta / alpha) + (((i * 4.0) / alpha) + ((beta - -2.0) / alpha))) / 2.0;
	} else {
		tmp = pow(pow(fma(((beta - alpha) / fma(2.0, i, ((alpha + beta) + 2.0))), ((alpha + beta) / (alpha + fma(2.0, i, beta))), 1.0), 3.0), 0.3333333333333333) / 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)) <= -0.999999)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(Float64(i * 4.0) / alpha) + Float64(Float64(beta - -2.0) / alpha))) / 2.0);
	else
		tmp = Float64(((fma(Float64(Float64(beta - alpha) / fma(2.0, i, Float64(Float64(alpha + beta) + 2.0))), Float64(Float64(alpha + beta) / Float64(alpha + fma(2.0, i, beta))), 1.0) ^ 3.0) ^ 0.3333333333333333) / 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], -0.999999], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(N[(i * 4.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $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)) < -0.999998999999999971

   1. Initial program 2.6%

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

   2. Simplified 14.3%

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

      [Start] 2.6	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 2.5	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      *-commutative [=>] 2.5	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      times-frac [=>] 14.3	\[ \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}} + 1}{2} \]
      fma-def [=>] 14.3	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}}{2} \]
      associate-+l+ [=>] 14.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      fma-def [=>] 14.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      associate-+l+ [=>] 14.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}, 1\right)}{2} \]
      +-commutative [=>] 14.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}, 1\right)}{2} \]
      fma-def [=>] 14.3	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}, 1\right)}{2} \]

   3. Taylor expanded in alpha around inf 91.8%

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

   4. Taylor expanded in i around 0 91.8%

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

   5. Simplified 91.8%

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

      [Start] 91.8	\[ \frac{\left(\frac{\beta}{\alpha} + 4 \cdot \frac{i}{\alpha}\right) - -1 \cdot \frac{\beta + 2}{\alpha}}{2} \]
      associate--l+ [=>] 91.8	\[ \frac{\color{blue}{\frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} - -1 \cdot \frac{\beta + 2}{\alpha}\right)}}{2} \]
      associate-*r/ [=>] 91.8	\[ \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\frac{4 \cdot i}{\alpha}} - -1 \cdot \frac{\beta + 2}{\alpha}\right)}{2} \]
      *-commutative [=>] 91.8	\[ \frac{\frac{\beta}{\alpha} + \left(\frac{\color{blue}{i \cdot 4}}{\alpha} - -1 \cdot \frac{\beta + 2}{\alpha}\right)}{2} \]
      associate-*r/ [=>] 91.8	\[ \frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} - \color{blue}{\frac{-1 \cdot \left(\beta + 2\right)}{\alpha}}\right)}{2} \]
      distribute-lft-in [=>] 91.8	\[ \frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} - \frac{\color{blue}{-1 \cdot \beta + -1 \cdot 2}}{\alpha}\right)}{2} \]
      neg-mul-1 [<=] 91.8	\[ \frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} - \frac{\color{blue}{\left(-\beta\right)} + -1 \cdot 2}{\alpha}\right)}{2} \]
      metadata-eval [=>] 91.8	\[ \frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} - \frac{\left(-\beta\right) + \color{blue}{-2}}{\alpha}\right)}{2} \]

   if -0.999998999999999971 < (/.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 80.3%

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

   2. Simplified 99.9%

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

      [Start] 80.3	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 80.3	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      *-commutative [=>] 80.3	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      times-frac [=>] 99.9	\[ \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}} + 1}{2} \]
      associate-+l+ [=>] 99.9	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2} \]
      fma-def [=>] 99.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} + 1}{2} \]
      fma-def [=>] 99.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} + 1}{2} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, \frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]
      Proof

      [Start] 99.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
      add-cbrt-cube [=>] 99.9	\[ \frac{\color{blue}{\sqrt[3]{\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)}}}{2} \]
      pow1/3 [=>] 99.9	\[ \frac{\color{blue}{{\left(\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)\right)}^{0.3333333333333333}}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} + \frac{\beta - -2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 2\right)}, \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} + \frac{\beta - -2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Accuracy	98.1%
Cost	16068

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} + \frac{\beta - -2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha - \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}}{2}\\ \end{array} \]

Alternative 3
Accuracy	97.6%
Cost	9540

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

Alternative 4
Accuracy	97.2%
Cost	3012

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

Alternative 5
Accuracy	87.3%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i} \cdot \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2\right) + \left(\beta + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	84.8%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	87.3%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2\right) + \left(\beta + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Accuracy	74.5%
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;i \leq 1.6 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;i \leq 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9
Accuracy	84.8%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{+191}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Accuracy	79.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Accuracy	76.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+214}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 12
Accuracy	78.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 13
Accuracy	72.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+46}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	61.5%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023146 
(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))