Octave 3.8, jcobi/3

Average Error: 3.7 → 0.3

Time: 15.2s

Precision: binary64

Cost: 1732

\[\alpha > -1 \land \beta > -1\]

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+77}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{t_1}}{t_1 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_0 + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+77)
     (/ (/ (* (+ alpha 1.0) (+ beta 1.0)) t_1) (* t_1 (+ alpha (+ beta 3.0))))
     (/ (/ (- alpha -1.0) t_0) (+ t_0 1.0)))))

C

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

↓

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+77) {
		tmp = (((alpha + 1.0) * (beta + 1.0)) / t_1) / (t_1 * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 1d+77) then
        tmp = (((alpha + 1.0d0) * (beta + 1.0d0)) / t_1) / (t_1 * (alpha + (beta + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / t_0) / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+77) {
		tmp = (((alpha + 1.0) * (beta + 1.0)) / t_1) / (t_1 * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)

↓

def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1e+77:
		tmp = (((alpha + 1.0) * (beta + 1.0)) / t_1) / (t_1 * (alpha + (beta + 3.0)))
	else:
		tmp = ((alpha - -1.0) / t_0) / (t_0 + 1.0)
	return tmp

Julia

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

↓

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1e+77)
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / t_1) / Float64(t_1 * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
end

↓

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1e+77)
		tmp = (((alpha + 1.0) * (beta + 1.0)) / t_1) / (t_1 * (alpha + (beta + 3.0)));
	else
		tmp = ((alpha - -1.0) / t_0) / (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+77], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999983e76

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
      Proof

      [Start] 0.1	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-38 [=>] 0.1	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-33 [=>] 0.1	\[ \frac{\frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-33 [=>] 0.1	\[ \frac{\frac{\color{blue}{\left(\left(1 + \alpha\right) + \beta\right)} + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-33 [<=] 0.1	\[ \frac{\frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-19 [=>] 0.1	\[ \frac{\frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-2 [=>] 0.1	\[ \frac{\frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right) \cdot \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-19 [=>] 0.1	\[ \frac{\frac{\color{blue}{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\color{blue}{\left(\alpha + 1\right)} \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \color{blue}{\left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-33 [<=] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-2 [=>] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      rational.json-simplify-33 [<=] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \color{blue}{\left(2 + \left(\alpha + \beta\right)\right)}\right)} \]

   if 9.99999999999999983e76 < beta

   1. Initial program 8.1

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

   2. Taylor expanded in beta around inf 0.5

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

   3. Simplified 0.5

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

      [Start] 0.5	\[ \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-1 [=>] 0.5	\[ \frac{\frac{\color{blue}{\alpha + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-12 [<=] 0.5	\[ \frac{\frac{\color{blue}{\alpha - -1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+77}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1604

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) + 1}{t_1 \cdot \left(t_1 \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_0 + 1}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1604

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 50000000000000:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_0 + 1}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	1220

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_0 + 1}\\ \end{array} \]

Alternative 5
Error	3.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1:\\ \;\;\;\;\frac{1 + \alpha}{\left(6 + 5 \cdot \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \end{array} \]

Alternative 6
Error	2.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \end{array} \]

Alternative 7
Error	1.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]

Alternative 8
Error	3.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{1 + \alpha}{\left(6 + 5 \cdot \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 9
Error	3.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.85:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \end{array} \]

Alternative 10
Error	3.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.92:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 11
Error	5.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.84:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 12
Error	3.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 13
Error	5.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 14
Error	35.4
Cost	320

\[0.08333333333333333 + -0.027777777777777776 \cdot \alpha \]

Alternative 15
Error	35.2
Cost	320

\[\frac{0.25}{3 + \alpha} \]

Alternative 16
Error	35.6
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023053 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))