Octave 3.8, jcobi/1

Average Error: 16.1 → 0.4

Time: 14.3s

Precision: binary64

Cost: 16964

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ alpha t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/
      (-
       (/ beta (+ beta (+ alpha 2.0)))
       (/
        (log
         (exp
          (-
           -1.0
           (*
            (/ alpha (/ t_0 alpha))
            (/ (/ alpha (- -2.0 (+ beta alpha))) t_0)))))
        (+ 1.0 (* t_1 (+ 1.0 t_1)))))
      2.0))))

C

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

↓

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

Fortran

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

Java

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

↓

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = alpha / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) - (Math.log(Math.exp((-1.0 - ((alpha / (t_0 / alpha)) * ((alpha / (-2.0 - (beta + alpha))) / t_0))))) / (1.0 + (t_1 * (1.0 + t_1))))) / 2.0;
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = alpha / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / (beta + (alpha + 2.0))) - (math.log(math.exp((-1.0 - ((alpha / (t_0 / alpha)) * ((alpha / (-2.0 - (beta + alpha))) / t_0))))) / (1.0 + (t_1 * (1.0 + t_1))))) / 2.0
	return tmp

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -1

   1. Initial program 60.6

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

   2. Simplified 60.6

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

      [Start] 60.6	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 60.6	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 0.0

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.6

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

   2. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.6	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.6

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

   4. Applied egg-rr 0.6

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

   5. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right) \cdot \frac{1}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      associate-*r/ [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\frac{\left(-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right) \cdot 1}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}}{2} \]
      *-commutative [<=] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{1 \cdot \left(-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      distribute-lft-in [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{1 \cdot -1 + 1 \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      metadata-eval [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{-1} + 1 \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      *-commutative [<=] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + \color{blue}{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot 1}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      *-rgt-identity [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + \color{blue}{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      associate-+r+ [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\color{blue}{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + 1\right) + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}}}}{2} \]
      +-commutative [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\color{blue}{\left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}}}{2} \]
      associate-+l+ [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\color{blue}{1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}}{2} \]
      *-rgt-identity [<=] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \left(\color{blue}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot 1} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      unpow2 [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot 1 + \color{blue}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}\right)}}{2} \]
      distribute-lft-in [<=] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \color{blue}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

   6. Applied egg-rr 0.6

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

   7. Applied egg-rr 0.6

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

   8. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\log \left(e^{-1 + \frac{\alpha}{\frac{\frac{\alpha + \left(\beta + 2\right)}{\alpha}}{\frac{-1}{-2 - \left(\alpha + \beta\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      associate-/r/ [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\log \left(e^{-1 + \color{blue}{\frac{\alpha}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}} \cdot \left(\frac{-1}{-2 - \left(\alpha + \beta\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      associate-*r/ [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\log \left(e^{-1 + \frac{\alpha}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}} \cdot \color{blue}{\frac{\frac{-1}{-2 - \left(\alpha + \beta\right)} \cdot \alpha}{\alpha + \left(\beta + 2\right)}}}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      associate-*l/ [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\log \left(e^{-1 + \frac{\alpha}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}} \cdot \frac{\color{blue}{\frac{-1 \cdot \alpha}{-2 - \left(\alpha + \beta\right)}}}{\alpha + \left(\beta + 2\right)}}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      neg-mul-1 [<=] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\log \left(e^{-1 + \frac{\alpha}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}} \cdot \frac{\frac{\color{blue}{-\alpha}}{-2 - \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 2\right)}}\right)}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	14404

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

Alternative 2
Error	0.4
Cost	1860

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

Alternative 3
Error	0.4
Cost	1476

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

Alternative 4
Error	9.3
Cost	717

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.5 \cdot 10^{+279} \lor \neg \left(\alpha \leq 1.8 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]

Alternative 5
Error	5.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	17.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]

Alternative 7
Error	18.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]

Alternative 8
Error	18.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	32.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023056 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))