Octave 3.8, jcobi/1

Average Accuracy: 74.8% → 99.6%

Time: 21.4s

Precision: binary64

Cost: 30788

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \left(\beta + \alpha\right) + 2\\ t_2 := \frac{\alpha}{t_1}\\ t_3 := \frac{\alpha}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{t_1} \leq -0.999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \frac{\frac{1 - {t_3}^{18}}{{t_3}^{12} + \left(1 + {t_3}^{6}\right)}}{\left(1 + t_2 \cdot \left(1 + t_2\right)\right) \cdot \left(1 + {t_2}^{3}\right)}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0)))
        (t_1 (+ (+ beta alpha) 2.0))
        (t_2 (/ alpha t_1))
        (t_3 (/ alpha t_0)))
   (if (<= (/ (- beta alpha) t_1) -0.999995)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/
      (+
       (/ beta t_0)
       (/
        (/ (- 1.0 (pow t_3 18.0)) (+ (pow t_3 12.0) (+ 1.0 (pow t_3 6.0))))
        (* (+ 1.0 (* t_2 (+ 1.0 t_2))) (+ 1.0 (pow t_2 3.0)))))
      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 = beta + (alpha + 2.0);
	double t_1 = (beta + alpha) + 2.0;
	double t_2 = alpha / t_1;
	double t_3 = alpha / t_0;
	double tmp;
	if (((beta - alpha) / t_1) <= -0.999995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (((1.0 - pow(t_3, 18.0)) / (pow(t_3, 12.0) + (1.0 + pow(t_3, 6.0)))) / ((1.0 + (t_2 * (1.0 + t_2))) * (1.0 + pow(t_2, 3.0))))) / 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = (beta + alpha) + 2.0d0
    t_2 = alpha / t_1
    t_3 = alpha / t_0
    if (((beta - alpha) / t_1) <= (-0.999995d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) + (((1.0d0 - (t_3 ** 18.0d0)) / ((t_3 ** 12.0d0) + (1.0d0 + (t_3 ** 6.0d0)))) / ((1.0d0 + (t_2 * (1.0d0 + t_2))) * (1.0d0 + (t_2 ** 3.0d0))))) / 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 = beta + (alpha + 2.0);
	double t_1 = (beta + alpha) + 2.0;
	double t_2 = alpha / t_1;
	double t_3 = alpha / t_0;
	double tmp;
	if (((beta - alpha) / t_1) <= -0.999995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (((1.0 - Math.pow(t_3, 18.0)) / (Math.pow(t_3, 12.0) + (1.0 + Math.pow(t_3, 6.0)))) / ((1.0 + (t_2 * (1.0 + t_2))) * (1.0 + Math.pow(t_2, 3.0))))) / 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 = beta + (alpha + 2.0)
	t_1 = (beta + alpha) + 2.0
	t_2 = alpha / t_1
	t_3 = alpha / t_0
	tmp = 0
	if ((beta - alpha) / t_1) <= -0.999995:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) + (((1.0 - math.pow(t_3, 18.0)) / (math.pow(t_3, 12.0) + (1.0 + math.pow(t_3, 6.0)))) / ((1.0 + (t_2 * (1.0 + t_2))) * (1.0 + math.pow(t_2, 3.0))))) / 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(beta + Float64(alpha + 2.0))
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	t_2 = Float64(alpha / t_1)
	t_3 = Float64(alpha / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_1) <= -0.999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(Float64(1.0 - (t_3 ^ 18.0)) / Float64((t_3 ^ 12.0) + Float64(1.0 + (t_3 ^ 6.0)))) / Float64(Float64(1.0 + Float64(t_2 * Float64(1.0 + t_2))) * Float64(1.0 + (t_2 ^ 3.0))))) / 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 = beta + (alpha + 2.0);
	t_1 = (beta + alpha) + 2.0;
	t_2 = alpha / t_1;
	t_3 = alpha / t_0;
	tmp = 0.0;
	if (((beta - alpha) / t_1) <= -0.999995)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) + (((1.0 - (t_3 ^ 18.0)) / ((t_3 ^ 12.0) + (1.0 + (t_3 ^ 6.0)))) / ((1.0 + (t_2 * (1.0 + t_2))) * (1.0 + (t_2 ^ 3.0))))) / 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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(alpha / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(alpha / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $MachinePrecision], -0.999995], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[t$95$3, 18.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$3, 12.0], $MachinePrecision] + N[(1.0 + N[Power[t$95$3, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(t$95$2 * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Power[t$95$2, 3.0], $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)) < -0.99999499999999997

   1. Initial program 7.4%

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

   2. Simplified 7.4%

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

      [Start] 7.4	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 7.4	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 98.7%

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

   if -0.99999499999999997 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\frac{-1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{6}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}}{2} \]
      Proof

      [Start] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{\left(1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}}{2} \]
      *-commutative [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{\color{blue}{\left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right) \cdot \left(1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)\right)}}}{2} \]
      associate-/r* [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\frac{\frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{1 + \color{blue}{\left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2} + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\color{blue}{\left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right) + \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}}{2} \]
      *-lft-identity [<=] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right) + \color{blue}{1 \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}}{2} \]
      metadata-eval [<=] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right) + \color{blue}{\left(--1\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}{2} \]
      cancel-sign-sub-inv [<=] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\color{blue}{\left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right) - -1 \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}}}{2} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{\frac{-1 + {\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{18}}{{\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{12} + \left({\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{6} + 1\right)}}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}{2} \]

   7. Simplified 99.9%

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{\frac{-1 + {\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{18}}{{\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{12} + \left(1 + {\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{6}\right)}}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}{2} \]
      Proof

      [Start] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{-1 + {\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{18}}{{\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{12} + \left({\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{6} + 1\right)}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}{2} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{-1 + {\left(\frac{\alpha}{\color{blue}{\left(\alpha + 2\right) + \beta}}\right)}^{18}}{{\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{12} + \left({\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{6} + 1\right)}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}{2} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{-1 + {\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{18}}{{\left(\frac{\alpha}{\color{blue}{\left(\alpha + 2\right) + \beta}}\right)}^{12} + \left({\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{6} + 1\right)}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{-1 + {\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{18}}{{\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{12} + \color{blue}{\left(1 + {\left(\frac{\alpha}{\alpha + \left(2 + \beta\right)}\right)}^{6}\right)}}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}{2} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{-1 + {\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{18}}{{\left(\frac{\alpha}{\left(\alpha + 2\right) + \beta}\right)}^{12} + \left(1 + {\left(\frac{\alpha}{\color{blue}{\left(\alpha + 2\right) + \beta}}\right)}^{6}\right)}}{\left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}\right)}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\frac{1 - {\left(\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}^{18}}{{\left(\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}^{12} + \left(1 + {\left(\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}^{6}\right)}}{\left(1 + \frac{\alpha}{\left(\beta + \alpha\right) + 2} \cdot \left(1 + \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	22404

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \frac{\alpha}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\log \left(e^{1 - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}\right)}{-1 + t_1 \cdot \left(-1 - t_1\right)}}{2}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	1860

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

Alternative 3
Accuracy	99.6%
Cost	1476

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

Alternative 4
Accuracy	90.9%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.08 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 7.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	91.0%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7 \cdot 10^{-12}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.08 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 6.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{2 + \frac{-2 - \alpha \cdot 2}{\beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	85.8%
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	90.9%
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.22 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 7.8 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Accuracy	85.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+68}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Accuracy	71.5%
Cost	580

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

Alternative 10
Accuracy	71.8%
Cost	580

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

Alternative 11
Accuracy	71.1%
Cost	196

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

Alternative 12
Accuracy	48.9%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023129 
(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))