Octave 3.8, jcobi/4

Average Accuracy: 15.6% → 84.9%

Time: 21.2s

Precision: binary64

Cost: 37380

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta \cdot 2 + \alpha \cdot 2\\ t_1 := {\left(\beta + \alpha\right)}^{2}\\ \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{\left(-2 \cdot \left(\left(\beta + \alpha\right) \cdot \left(0.0625 \cdot t_0 - \left(\beta + \alpha\right) \cdot 0.125\right)\right) + -0.00390625 \cdot \left(t_1 \cdot 16 + \left(t_1 \cdot 4 + 4 \cdot \left(-1 + t_1\right)\right)\right)\right) + 0.0625 \cdot \left(\beta \cdot \alpha + t_1\right)}{{i}^{2}} + \left(0.0625 + 0.0625 \cdot \frac{t_0}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

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 (+ (* beta 2.0) (* alpha 2.0))) (t_1 (pow (+ beta alpha) 2.0)))
   (if (<= beta 2.7e+134)
     (-
      (+
       (/
        (+
         (+
          (*
           -2.0
           (* (+ beta alpha) (- (* 0.0625 t_0) (* (+ beta alpha) 0.125))))
          (*
           -0.00390625
           (+ (* t_1 16.0) (+ (* t_1 4.0) (* 4.0 (+ -1.0 t_1))))))
         (* 0.0625 (+ (* beta alpha) t_1)))
        (pow i 2.0))
       (+ 0.0625 (* 0.0625 (/ t_0 i))))
      (* 0.125 (/ (+ beta alpha) i)))
     (* (/ (+ alpha i) beta) (/ i beta)))))

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 = (beta * 2.0) + (alpha * 2.0);
	double t_1 = pow((beta + alpha), 2.0);
	double tmp;
	if (beta <= 2.7e+134) {
		tmp = (((((-2.0 * ((beta + alpha) * ((0.0625 * t_0) - ((beta + alpha) * 0.125)))) + (-0.00390625 * ((t_1 * 16.0) + ((t_1 * 4.0) + (4.0 * (-1.0 + t_1)))))) + (0.0625 * ((beta * alpha) + t_1))) / pow(i, 2.0)) + (0.0625 + (0.0625 * (t_0 / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0d0 * i)) * ((alpha + beta) + (2.0d0 * i)))) / ((((alpha + beta) + (2.0d0 * i)) * ((alpha + beta) + (2.0d0 * i))) - 1.0d0)
end function

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta * 2.0d0) + (alpha * 2.0d0)
    t_1 = (beta + alpha) ** 2.0d0
    if (beta <= 2.7d+134) then
        tmp = ((((((-2.0d0) * ((beta + alpha) * ((0.0625d0 * t_0) - ((beta + alpha) * 0.125d0)))) + ((-0.00390625d0) * ((t_1 * 16.0d0) + ((t_1 * 4.0d0) + (4.0d0 * ((-1.0d0) + t_1)))))) + (0.0625d0 * ((beta * alpha) + t_1))) / (i ** 2.0d0)) + (0.0625d0 + (0.0625d0 * (t_0 / i)))) - (0.125d0 * ((beta + alpha) / i))
    else
        tmp = ((alpha + i) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static 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);
}

↓

public static double code(double alpha, double beta, double i) {
	double t_0 = (beta * 2.0) + (alpha * 2.0);
	double t_1 = Math.pow((beta + alpha), 2.0);
	double tmp;
	if (beta <= 2.7e+134) {
		tmp = (((((-2.0 * ((beta + alpha) * ((0.0625 * t_0) - ((beta + alpha) * 0.125)))) + (-0.00390625 * ((t_1 * 16.0) + ((t_1 * 4.0) + (4.0 * (-1.0 + t_1)))))) + (0.0625 * ((beta * alpha) + t_1))) / Math.pow(i, 2.0)) + (0.0625 + (0.0625 * (t_0 / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, 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)

↓

def code(alpha, beta, i):
	t_0 = (beta * 2.0) + (alpha * 2.0)
	t_1 = math.pow((beta + alpha), 2.0)
	tmp = 0
	if beta <= 2.7e+134:
		tmp = (((((-2.0 * ((beta + alpha) * ((0.0625 * t_0) - ((beta + alpha) * 0.125)))) + (-0.00390625 * ((t_1 * 16.0) + ((t_1 * 4.0) + (4.0 * (-1.0 + t_1)))))) + (0.0625 * ((beta * alpha) + t_1))) / math.pow(i, 2.0)) + (0.0625 + (0.0625 * (t_0 / i)))) - (0.125 * ((beta + alpha) / i))
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	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 = Float64(Float64(beta * 2.0) + Float64(alpha * 2.0))
	t_1 = Float64(beta + alpha) ^ 2.0
	tmp = 0.0
	if (beta <= 2.7e+134)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(beta + alpha) * Float64(Float64(0.0625 * t_0) - Float64(Float64(beta + alpha) * 0.125)))) + Float64(-0.00390625 * Float64(Float64(t_1 * 16.0) + Float64(Float64(t_1 * 4.0) + Float64(4.0 * Float64(-1.0 + t_1)))))) + Float64(0.0625 * Float64(Float64(beta * alpha) + t_1))) / (i ^ 2.0)) + Float64(0.0625 + Float64(0.0625 * Float64(t_0 / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((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);
end

↓

function tmp_2 = code(alpha, beta, i)
	t_0 = (beta * 2.0) + (alpha * 2.0);
	t_1 = (beta + alpha) ^ 2.0;
	tmp = 0.0;
	if (beta <= 2.7e+134)
		tmp = (((((-2.0 * ((beta + alpha) * ((0.0625 * t_0) - ((beta + alpha) * 0.125)))) + (-0.00390625 * ((t_1 * 16.0) + ((t_1 * 4.0) + (4.0 * (-1.0 + t_1)))))) + (0.0625 * ((beta * alpha) + t_1))) / (i ^ 2.0)) + (0.0625 + (0.0625 * (t_0 / i)))) - (0.125 * ((beta + alpha) / i));
	else
		tmp = ((alpha + i) / beta) * (i / beta);
	end
	tmp_2 = 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[(N[(beta * 2.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 2.7e+134], N[(N[(N[(N[(N[(N[(-2.0 * N[(N[(beta + alpha), $MachinePrecision] * N[(N[(0.0625 * t$95$0), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.00390625 * N[(N[(t$95$1 * 16.0), $MachinePrecision] + N[(N[(t$95$1 * 4.0), $MachinePrecision] + N[(4.0 * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[(N[(beta * alpha), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(0.0625 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7e134

   1. Initial program 23.2%

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

   2. Simplified 50.4%

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

      [Start] 23.2	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 21.2	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      associate-*l* [=>] 21.1	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      times-frac [=>] 32.3	\[ \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Taylor expanded in i around -inf 91.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{0.0625 \cdot \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) - \left(-2 \cdot \left(\left(0.0625 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.125 \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)\right) + -0.00390625 \cdot \left(16 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + 4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right)\right)\right)\right)}{{i}^{2}} + \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   if 2.7e134 < beta

   1. Initial program 0.3%

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

   2. Simplified 0.0%

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

      [Start] 0.3	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{\left(i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 0.0	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 0.0	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Taylor expanded in beta around inf 26.1%

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

   4. Simplified 28.0%

      \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{\beta \cdot \beta}{i}}} \]
      Proof

      [Start] 26.1	\[ \frac{\left(i + \alpha\right) \cdot i}{{\beta}^{2}} \]
      associate-/l* [=>] 28.0	\[ \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]
      unpow2 [=>] 28.0	\[ \frac{i + \alpha}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}} \]
      Proof

      [Start] 28.0	\[ \frac{i + \alpha}{\frac{\beta \cdot \beta}{i}} \]
      associate-/l* [=>] 47.7	\[ \frac{i + \alpha}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]
      associate-/r/ [=>] 71.2	\[ \color{blue}{\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{\left(-2 \cdot \left(\left(\beta + \alpha\right) \cdot \left(0.0625 \cdot \left(\beta \cdot 2 + \alpha \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.125\right)\right) + -0.00390625 \cdot \left({\left(\beta + \alpha\right)}^{2} \cdot 16 + \left({\left(\beta + \alpha\right)}^{2} \cdot 4 + 4 \cdot \left(-1 + {\left(\beta + \alpha\right)}^{2}\right)\right)\right)\right) + 0.0625 \cdot \left(\beta \cdot \alpha + {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}} + \left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 2
Accuracy	75.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+190}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 3
Accuracy	82.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Accuracy	82.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 5
Accuracy	71.0%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023153 
(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)))