Octave 3.8, jcobi/1

Average Error: 16.3 → 0.1

Time: 12.9s

Precision: binary64

Cost: 14660

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} - \frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(\beta + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{-1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}}\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
   (/
    (-
     (/ (+ beta (+ beta 2.0)) alpha)
     (* (/ (/ (+ 2.0 (+ beta beta)) alpha) alpha) (+ beta 2.0)))
    2.0)
   (/
    (-
     (/ beta (+ beta (+ alpha 2.0)))
     (log (exp (+ -1.0 (/ alpha (+ 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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = (((beta + (beta + 2.0)) / alpha) - ((((2.0 + (beta + beta)) / alpha) / alpha) * (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) - log(exp((-1.0 + (alpha / (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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9998d0)) then
        tmp = (((beta + (beta + 2.0d0)) / alpha) - ((((2.0d0 + (beta + beta)) / alpha) / alpha) * (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta / (beta + (alpha + 2.0d0))) - log(exp(((-1.0d0) + (alpha / (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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = (((beta + (beta + 2.0)) / alpha) - ((((2.0 + (beta + beta)) / alpha) / alpha) * (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) - Math.log(Math.exp((-1.0 + (alpha / (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):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998:
		tmp = (((beta + (beta + 2.0)) / alpha) - ((((2.0 + (beta + beta)) / alpha) / alpha) * (beta + 2.0))) / 2.0
	else:
		tmp = ((beta / (beta + (alpha + 2.0))) - math.log(math.exp((-1.0 + (alpha / (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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9998)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) - Float64(Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) / alpha) * Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) - log(exp(Float64(-1.0 + Float64(alpha / 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)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998)
		tmp = (((beta + (beta + 2.0)) / alpha) - ((((2.0 + (beta + beta)) / alpha) / alpha) * (beta + 2.0))) / 2.0;
	else
		tmp = ((beta / (beta + (alpha + 2.0))) - log(exp((-1.0 + (alpha / (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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / alpha), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[Exp[N[(-1.0 + N[(alpha / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $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)) < -0.99980000000000002

   1. Initial program 59.4

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

   2. Simplified 59.4

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

      [Start] 59.4	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.4	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 3.0

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

   4. Simplified 3.0

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

      [Start] 3.0	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      distribute-lft-out [=>] 3.0	\[ \frac{\color{blue}{-1 \cdot \left(\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      mul-1-neg [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \color{blue}{\left(2 + \beta\right)}}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\color{blue}{\left(2 + \beta\right)}}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \left(2 + \beta\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   5. Applied egg-rr 45.9

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

   6. Taylor expanded in alpha around 0 3.0

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

   7. Simplified 0.2

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

      [Start] 3.0	\[ \frac{-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + \frac{\left(2 + 2 \cdot \beta\right) \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 3.0	\[ \frac{-1 \cdot \color{blue}{\left(\frac{\left(2 + 2 \cdot \beta\right) \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)}}{2} \]
      associate-/l* [=>] 0.2	\[ \frac{-1 \cdot \left(\color{blue}{\frac{2 + 2 \cdot \beta}{\frac{{\alpha}^{2}}{\beta + 2}}} + -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)}{2} \]
      count-2 [<=] 0.2	\[ \frac{-1 \cdot \left(\frac{2 + \color{blue}{\left(\beta + \beta\right)}}{\frac{{\alpha}^{2}}{\beta + 2}} + -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)}{2} \]
      unpow2 [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{2 + \left(\beta + \beta\right)}{\frac{\color{blue}{\alpha \cdot \alpha}}{\beta + 2}} + -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)}{2} \]
      associate-/r/ [=>] 0.2	\[ \frac{-1 \cdot \left(\color{blue}{\frac{2 + \left(\beta + \beta\right)}{\alpha \cdot \alpha} \cdot \left(\beta + 2\right)} + -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)}{2} \]
      associate-/r* [=>] 0.2	\[ \frac{-1 \cdot \left(\color{blue}{\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha}} \cdot \left(\beta + 2\right) + -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)}{2} \]
      +-commutative [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \color{blue}{\left(2 + \beta\right)} + -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)}{2} \]
      associate-*r/ [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \color{blue}{\frac{-1 \cdot \left(2 + 2 \cdot \beta\right)}{\alpha}}\right)}{2} \]
      count-2 [<=] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{-1 \cdot \left(2 + \color{blue}{\left(\beta + \beta\right)}\right)}{\alpha}\right)}{2} \]
      neg-mul-1 [<=] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\color{blue}{-\left(2 + \left(\beta + \beta\right)\right)}}{\alpha}\right)}{2} \]
      neg-sub0 [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\color{blue}{0 - \left(2 + \left(\beta + \beta\right)\right)}}{\alpha}\right)}{2} \]
      associate-+r+ [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{0 - \color{blue}{\left(\left(2 + \beta\right) + \beta\right)}}{\alpha}\right)}{2} \]
      +-commutative [<=] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{0 - \left(\color{blue}{\left(\beta + 2\right)} + \beta\right)}{\alpha}\right)}{2} \]
      associate--r+ [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\color{blue}{\left(0 - \left(\beta + 2\right)\right) - \beta}}{\alpha}\right)}{2} \]
      neg-sub0 [<=] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\color{blue}{\left(-\left(\beta + 2\right)\right)} - \beta}{\alpha}\right)}{2} \]
      +-commutative [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\left(-\color{blue}{\left(2 + \beta\right)}\right) - \beta}{\alpha}\right)}{2} \]
      distribute-neg-in [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\color{blue}{\left(\left(-2\right) + \left(-\beta\right)\right)} - \beta}{\alpha}\right)}{2} \]
      metadata-eval [=>] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\left(\color{blue}{-2} + \left(-\beta\right)\right) - \beta}{\alpha}\right)}{2} \]
      sub-neg [<=] 0.2	\[ \frac{-1 \cdot \left(\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \frac{\color{blue}{\left(-2 - \beta\right)} - \beta}{\alpha}\right)}{2} \]

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.0

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

   4. Applied egg-rr 0.0

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} - \frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(\beta + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{-1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}}\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	2116

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} - \frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{\alpha} \cdot \left(\beta + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \left(-1 + \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} + \frac{\frac{-4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \left(-1 + \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1604

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	1604

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} + \frac{\frac{-4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 5
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 6
Error	4.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.7:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	7.5
Cost	708

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

Alternative 8
Error	4.5
Cost	708

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

Alternative 9
Error	18.2
Cost	580

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

Alternative 10
Error	18.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.98:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

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	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023053 
(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))