Octave 3.8, jcobi/2

Average Accuracy: 62.7% → 97.8%

Time: 38.5s

Precision: binary64

Cost: 35140

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99998:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\beta + \left(2 + \mathsf{fma}\left(2, i, \alpha\right)\right)}, 1\right)\right)}}{2}\\ \end{array} \]

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)) -0.99998)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)
     (/
      (exp
       (log
        (fma
         (/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))
         (/ (- beta alpha) (+ beta (+ 2.0 (fma 2.0 i alpha))))
         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)) <= -0.99998) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = exp(log(fma(((alpha + beta) / (beta + fma(2.0, i, alpha))), ((beta - alpha) / (beta + (2.0 + fma(2.0, i, alpha)))), 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)) <= -0.99998)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(exp(log(fma(Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha))), Float64(Float64(beta - alpha) / Float64(beta + Float64(2.0 + fma(2.0, i, alpha)))), 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], -0.99998], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -0.99997999999999998

   1. Initial program 2.9%

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

   2. Simplified 2.9%

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

      [Start] 2.9	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 2.9	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      associate-+l+ [=>] 2.9	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      associate-+l+ [=>] 2.9	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Applied egg-rr 3.2%

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

      [Start] 2.9	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2} \]
      div-inv [=>] 3.0	\[ \frac{\color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]
      fma-def [=>] 3.2	\[ \frac{\color{blue}{\mathsf{fma}\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right), \frac{1}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}, 1\right)}}{2} \]
      +-commutative [=>] 3.2	\[ \frac{\mathsf{fma}\left(\color{blue}{\left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right), \frac{1}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}, 1\right)}{2} \]
      difference-of-squares [<=] 3.2	\[ \frac{\mathsf{fma}\left(\color{blue}{\beta \cdot \beta - \alpha \cdot \alpha}, \frac{1}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}, 1\right)}{2} \]
      associate-+l+ [=>] 3.2	\[ \frac{\mathsf{fma}\left(\beta \cdot \beta - \alpha \cdot \alpha, \frac{1}{\color{blue}{\left(\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}, 1\right)}{2} \]
      fma-def [=>] 3.2	\[ \frac{\mathsf{fma}\left(\beta \cdot \beta - \alpha \cdot \alpha, \frac{1}{\left(\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}, 1\right)}{2} \]
      +-commutative [=>] 3.2	\[ \frac{\mathsf{fma}\left(\beta \cdot \beta - \alpha \cdot \alpha, \frac{1}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \left(\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}\right)}, 1\right)}{2} \]
      fma-def [=>] 3.2	\[ \frac{\mathsf{fma}\left(\beta \cdot \beta - \alpha \cdot \alpha, \frac{1}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \left(\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}\right)}, 1\right)}{2} \]

   4. Simplified 3.2%

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

      [Start] 3.2	\[ \frac{\mathsf{fma}\left(\beta \cdot \beta - \alpha \cdot \alpha, \frac{1}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right)}, 1\right)}{2} \]
      +-commutative [=>] 3.2	\[ \frac{\mathsf{fma}\left(\beta \cdot \beta - \alpha \cdot \alpha, \frac{1}{\color{blue}{\left(\left(\beta + \mathsf{fma}\left(2, i, 2\right)\right) + \alpha\right)} \cdot \left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right)}, 1\right)}{2} \]

   5. Taylor expanded in alpha around inf 90.4%

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

   if -0.99997999999999998 < (/.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 80.0%

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

   2. Simplified 99.9%

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

      [Start] 80.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 80.0	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 99.9	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 99.9	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

   3. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2} \]
      add-exp-log [=>] 99.9	\[ \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)\right)}}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\beta + \mathsf{fma}\left(2, i, 2\right)\right) + \alpha}}, 1\right)\right)}}{2} \]
      associate-+l+ [=>] 99.9	\[ \frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\beta + \left(\mathsf{fma}\left(2, i, 2\right) + \alpha\right)}}, 1\right)\right)}}{2} \]
      fma-udef [=>] 99.9	\[ \frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\beta + \left(\color{blue}{\left(2 \cdot i + 2\right)} + \alpha\right)}, 1\right)\right)}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\beta + \left(\color{blue}{\left(2 + 2 \cdot i\right)} + \alpha\right)}, 1\right)\right)}}{2} \]
      associate-+l+ [=>] 99.9	\[ \frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\beta + \color{blue}{\left(2 + \left(2 \cdot i + \alpha\right)\right)}}, 1\right)\right)}}{2} \]
      fma-udef [<=] 99.9	\[ \frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\beta + \left(2 + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}\right)}, 1\right)\right)}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99998:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\beta + \left(2 + \mathsf{fma}\left(2, i, \alpha\right)\right)}, 1\right)\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.8%
Cost	9796

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.99998:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1}}{2}\\ \end{array} \]

Alternative 2
Accuracy	96.2%
Cost	2628

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

Alternative 3
Accuracy	77.4%
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+42} \lor \neg \left(\alpha \leq 10^{+75}\right) \land \alpha \leq 1.25 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Accuracy	79.8%
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 5.3 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.95 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	83.3%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	74.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.65 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	72.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	61.8%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023133 
(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))