Octave 3.8, jcobi/1

Average Error: 16.1 → 0.2

Time: 14.3s

Precision: binary64

Cost: 8580

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

Math

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

↓

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.999999999999:\\ \;\;\;\;\left(\left(\beta + 2\right) \cdot \frac{-1 - \beta}{{\alpha}^{2}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}\right) - \frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{\alpha}{t_0} + \frac{\beta}{t_0}\right)\right) \cdot 1 - -2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- -2.0 (+ alpha beta))))
   (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.999999999999)
     (-
      (+
       (* (+ beta 2.0) (/ (- -1.0 beta) (pow alpha 2.0)))
       (/ (+ beta (- beta -2.0)) alpha))
      (/ (+ beta 1.0) alpha))
     (/
      (-
       (* (+ 1.0 (+ (/ alpha t_0) (/ beta t_0))) 1.0)
       (* -2.0 (/ beta (+ alpha (+ beta 2.0)))))
      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 = -2.0 - (alpha + beta);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.999999999999) {
		tmp = (((beta + 2.0) * ((-1.0 - beta) / pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) - ((beta + 1.0) / alpha);
	} else {
		tmp = (((1.0 + ((alpha / t_0) + (beta / t_0))) * 1.0) - (-2.0 * (beta / (alpha + (beta + 2.0))))) / 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) :: tmp
    t_0 = (-2.0d0) - (alpha + beta)
    if (((beta - alpha) / ((alpha + beta) + 2.0d0)) <= (-0.999999999999d0)) then
        tmp = (((beta + 2.0d0) * (((-1.0d0) - beta) / (alpha ** 2.0d0))) + ((beta + (beta - (-2.0d0))) / alpha)) - ((beta + 1.0d0) / alpha)
    else
        tmp = (((1.0d0 + ((alpha / t_0) + (beta / t_0))) * 1.0d0) - ((-2.0d0) * (beta / (alpha + (beta + 2.0d0))))) / 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 = -2.0 - (alpha + beta);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.999999999999) {
		tmp = (((beta + 2.0) * ((-1.0 - beta) / Math.pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) - ((beta + 1.0) / alpha);
	} else {
		tmp = (((1.0 + ((alpha / t_0) + (beta / t_0))) * 1.0) - (-2.0 * (beta / (alpha + (beta + 2.0))))) / 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 = -2.0 - (alpha + beta)
	tmp = 0
	if ((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.999999999999:
		tmp = (((beta + 2.0) * ((-1.0 - beta) / math.pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) - ((beta + 1.0) / alpha)
	else:
		tmp = (((1.0 + ((alpha / t_0) + (beta / t_0))) * 1.0) - (-2.0 * (beta / (alpha + (beta + 2.0))))) / 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(-2.0 - Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.999999999999)
		tmp = Float64(Float64(Float64(Float64(beta + 2.0) * Float64(Float64(-1.0 - beta) / (alpha ^ 2.0))) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) - Float64(Float64(beta + 1.0) / alpha));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(alpha / t_0) + Float64(beta / t_0))) * 1.0) - Float64(-2.0 * Float64(beta / Float64(alpha + Float64(beta + 2.0))))) / 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 = -2.0 - (alpha + beta);
	tmp = 0.0;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.999999999999)
		tmp = (((beta + 2.0) * ((-1.0 - beta) / (alpha ^ 2.0))) + ((beta + (beta - -2.0)) / alpha)) - ((beta + 1.0) / alpha);
	else
		tmp = (((1.0 + ((alpha / t_0) + (beta / t_0))) * 1.0) - (-2.0 * (beta / (alpha + (beta + 2.0))))) / 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[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999999999], N[(N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(-1.0 - beta), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(N[(alpha / t$95$0), $MachinePrecision] + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(-2.0 * N[(beta / N[(alpha + N[(beta + 2.0), $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)) < -0.999999999999000022

   1. Initial program 60.4

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

   2. Simplified 60.4

      \[\leadsto \color{blue}{0.5 - \frac{0.5}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha - \beta\right)} \]
      Proof

      [Start] 60.4	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational.json-simplify-78 [=>] 60.4	\[ \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - -1}}{2} \]
      rational.json-simplify-23 [=>] 60.4	\[ \color{blue}{\frac{-1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{-2}} \]
      rational.json-simplify-31 [=>] 60.4	\[ \color{blue}{\frac{-1}{-2} - \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{-2}} \]
      metadata-eval [=>] 60.4	\[ \frac{-1}{\color{blue}{-2}} - \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{-2} \]
      metadata-eval [=>] 60.4	\[ \color{blue}{0.5} - \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{-2} \]
      rational.json-simplify-7 [=>] 60.4	\[ 0.5 - \color{blue}{\frac{\frac{\beta - \alpha}{-2}}{\left(\alpha + \beta\right) + 2}} \]
      rational.json-simplify-23 [<=] 60.4	\[ 0.5 - \frac{\color{blue}{\frac{\alpha - \beta}{2}}}{\left(\alpha + \beta\right) + 2} \]
      rational.json-simplify-31 [=>] 60.4	\[ 0.5 - \frac{\color{blue}{\frac{\alpha}{2} - \frac{\beta}{2}}}{\left(\alpha + \beta\right) + 2} \]
      rational.json-simplify-31 [=>] 60.3	\[ 0.5 - \color{blue}{\left(\frac{\frac{\alpha}{2}}{\left(\alpha + \beta\right) + 2} - \frac{\frac{\beta}{2}}{\left(\alpha + \beta\right) + 2}\right)} \]

   3. Taylor expanded in alpha around inf 3.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{\left(\beta + 2\right) \cdot \left(-1 \cdot \beta - \left(\beta + 2\right)\right)}{{\alpha}^{2}} + -0.5 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\left(\left(-\beta\right) - \left(\beta + 2\right)\right) \cdot -0.5}{\alpha} + 0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\left(-\beta\right) - \left(\beta + 2\right)}{{\alpha}^{2}}\right)} \]
      Proof

      [Start] 3.0	\[ 0.5 \cdot \frac{\left(\beta + 2\right) \cdot \left(-1 \cdot \beta - \left(\beta + 2\right)\right)}{{\alpha}^{2}} + -0.5 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} \]
      rational.json-simplify-41 [=>] 3.0	\[ \color{blue}{-0.5 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + 0.5 \cdot \frac{\left(\beta + 2\right) \cdot \left(-1 \cdot \beta - \left(\beta + 2\right)\right)}{{\alpha}^{2}}} \]
      rational.json-simplify-39 [=>] 3.0	\[ -0.5 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + 0.5 \cdot \frac{\color{blue}{\left(-1 \cdot \beta - \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)}}{{\alpha}^{2}} \]
      rational.json-simplify-20 [=>] 3.0	\[ \color{blue}{\frac{\left(-1 \cdot \beta - \left(\beta + 2\right)\right) \cdot -0.5}{\alpha}} + 0.5 \cdot \frac{\left(-1 \cdot \beta - \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)}{{\alpha}^{2}} \]
      rational.json-simplify-39 [=>] 3.0	\[ \frac{\left(\color{blue}{\beta \cdot -1} - \left(\beta + 2\right)\right) \cdot -0.5}{\alpha} + 0.5 \cdot \frac{\left(-1 \cdot \beta - \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)}{{\alpha}^{2}} \]
      rational.json-simplify-71 [<=] 3.0	\[ \frac{\left(\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)\right) \cdot -0.5}{\alpha} + 0.5 \cdot \frac{\left(-1 \cdot \beta - \left(\beta + 2\right)\right) \cdot \left(\beta + 2\right)}{{\alpha}^{2}} \]
      rational.json-simplify-19 [=>] 0.1	\[ \frac{\left(\left(-\beta\right) - \left(\beta + 2\right)\right) \cdot -0.5}{\alpha} + 0.5 \cdot \color{blue}{\left(\left(\beta + 2\right) \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{{\alpha}^{2}}\right)} \]
      rational.json-simplify-39 [=>] 0.1	\[ \frac{\left(\left(-\beta\right) - \left(\beta + 2\right)\right) \cdot -0.5}{\alpha} + 0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\color{blue}{\beta \cdot -1} - \left(\beta + 2\right)}{{\alpha}^{2}}\right) \]
      rational.json-simplify-71 [<=] 0.1	\[ \frac{\left(\left(-\beta\right) - \left(\beta + 2\right)\right) \cdot -0.5}{\alpha} + 0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)}{{\alpha}^{2}}\right) \]

   5. Applied egg-rr 0.1

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

   if -0.999999999999000022 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.3

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

   2. Simplified 0.3

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

      [Start] 0.3	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational.json-simplify-41 [=>] 0.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.3

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

   5. Simplified 0.2

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

      [Start] 0.3	\[ \frac{1 \cdot \left(\left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} - -1\right)\right) - -2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-34 [=>] 0.3	\[ \frac{\color{blue}{\left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} - -1\right)\right) \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      rational.json-simplify-21 [=>] 0.3	\[ \frac{\color{blue}{\left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} + \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} - -1\right)\right)} \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-78 [<=] 0.3	\[ \frac{\left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} + \color{blue}{\left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + 1\right)}\right) \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-41 [=>] 0.3	\[ \frac{\left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} + \color{blue}{\left(1 + \frac{\alpha}{-2 - \left(\alpha + \beta\right)}\right)}\right) \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-11 [=>] 0.2	\[ \frac{\color{blue}{\left(1 + \left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} + \frac{\alpha}{-2 - \left(\alpha + \beta\right)}\right)\right)} \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-65 [<=] 0.2	\[ \frac{\left(1 + \left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} + \color{blue}{\left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + 0\right)}\right)\right) \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-11 [<=] 0.2	\[ \frac{\left(1 + \color{blue}{\left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \left(\frac{\beta}{-2 - \left(\alpha + \beta\right)} + 0\right)\right)}\right) \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-65 [=>] 0.2	\[ \frac{\left(1 + \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \color{blue}{\frac{\beta}{-2 - \left(\alpha + \beta\right)}}\right)\right) \cdot 1 - 1 \cdot \left(-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}{2} \]
      rational.json-simplify-39 [=>] 0.2	\[ \frac{\left(1 + \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \frac{\beta}{-2 - \left(\alpha + \beta\right)}\right)\right) \cdot 1 - 1 \cdot \color{blue}{\left(\frac{\beta}{\alpha + \left(\beta + 2\right)} \cdot -2\right)}}{2} \]
      rational.json-simplify-3 [=>] 0.2	\[ \frac{\left(1 + \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \frac{\beta}{-2 - \left(\alpha + \beta\right)}\right)\right) \cdot 1 - \color{blue}{\frac{\beta}{\alpha + \left(\beta + 2\right)} \cdot \left(1 \cdot -2\right)}}{2} \]
      metadata-eval [=>] 0.2	\[ \frac{\left(1 + \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \frac{\beta}{-2 - \left(\alpha + \beta\right)}\right)\right) \cdot 1 - \frac{\beta}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{-2}}{2} \]
      rational.json-simplify-39 [<=] 0.2	\[ \frac{\left(1 + \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \frac{\beta}{-2 - \left(\alpha + \beta\right)}\right)\right) \cdot 1 - \color{blue}{-2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.999999999999:\\ \;\;\;\;\left(\left(\beta + 2\right) \cdot \frac{-1 - \beta}{{\alpha}^{2}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}\right) - \frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{\alpha}{-2 - \left(\alpha + \beta\right)} + \frac{\beta}{-2 - \left(\alpha + \beta\right)}\right)\right) \cdot 1 - -2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	2628

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.999999999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{\alpha}{t_0} + \frac{\beta}{t_0}\right)\right) \cdot 1 - -2 \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \frac{\alpha - 0.5 \cdot \left(\beta + \alpha\right)}{\beta + \left(2 + \alpha\right)}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \frac{\alpha - 0.5 \cdot \left(\beta + \alpha\right)}{\beta + \left(2 + \alpha\right)}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 5
Error	3.6
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;0.5 - \frac{0.5}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha - \beta\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 6
Error	16.4
Cost	844

\[\begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{-76}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 7
Error	16.6
Cost	716

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.65 \cdot 10^{-83}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{-76}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 7.2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 8
Error	14.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.38:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + \alpha}{\beta} - -1}\\ \end{array} \]

Alternative 9
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 310000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 10
Error	18.3
Cost	452

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

Alternative 11
Error	18.4
Cost	196

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

Alternative 12
Error	40.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(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))