Octave 3.8, jcobi/1

Average Error: 16.0 → 0.3

Time: 18.3s

Precision: binary64

Cost: 16836

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha}{t_0}\\ t_2 := \frac{t_0}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\frac{-1}{\frac{-1 - {t_2}^{-3}}{1 - {t_2}^{-6}}}}{1 + t_1 \cdot \left(1 + t_1\right)}}{2}\\ \end{array} \]

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)) (t_2 (/ t_0 alpha)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
     (/ (log1p (/ (+ 2.0 (+ beta beta)) alpha)) 2.0)
     (/
      (+
       (/ beta (+ beta (+ alpha 2.0)))
       (/
        (/ -1.0 (/ (- -1.0 (pow t_2 -3.0)) (- 1.0 (pow t_2 -6.0))))
        (+ 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 t_2 = t_0 / alpha;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995) {
		tmp = log1p(((2.0 + (beta + beta)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((-1.0 / ((-1.0 - pow(t_2, -3.0)) / (1.0 - pow(t_2, -6.0)))) / (1.0 + (t_1 * (1.0 + t_1))))) / 2.0;
	}
	return tmp;
}

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 t_2 = t_0 / alpha;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995) {
		tmp = Math.log1p(((2.0 + (beta + beta)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((-1.0 / ((-1.0 - Math.pow(t_2, -3.0)) / (1.0 - Math.pow(t_2, -6.0)))) / (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
	t_2 = t_0 / alpha
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995:
		tmp = math.log1p(((2.0 + (beta + beta)) / alpha)) / 2.0
	else:
		tmp = ((beta / (beta + (alpha + 2.0))) + ((-1.0 / ((-1.0 - math.pow(t_2, -3.0)) / (1.0 - math.pow(t_2, -6.0)))) / (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)
	t_2 = Float64(t_0 / alpha)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.995)
		tmp = Float64(log1p(Float64(Float64(2.0 + Float64(beta + beta)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) + Float64(Float64(-1.0 / Float64(Float64(-1.0 - (t_2 ^ -3.0)) / Float64(1.0 - (t_2 ^ -6.0)))) / Float64(1.0 + Float64(t_1 * Float64(1.0 + t_1))))) / 2.0);
	end
	return 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]}, Block[{t$95$2 = N[(t$95$0 / alpha), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.995], N[(N[Log[1 + N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[(N[(-1.0 - N[Power[t$95$2, -3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$2, -6.0], $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)) < -0.994999999999999996

   1. Initial program 58.7

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

   2. Simplified 58.7

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

      [Start] 58.7	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 58.7	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 58.7

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

   4. Taylor expanded in alpha around -inf 59.0

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

   5. Applied egg-rr 1.0

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right) + 0}}{2} \]

   6. Simplified 1.0

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}}{2} \]
      Proof

      [Start] 1.0	\[ \frac{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right) + 0}{2} \]
      +-rgt-identity [=>] 1.0	\[ \frac{\color{blue}{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}}{2} \]

   if -0.994999999999999996 < (/.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

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

   5. Simplified 0.0

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

      [Start] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right) \cdot \frac{1}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      associate-*r/ [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\frac{\left(-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right) \cdot 1}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}}{2} \]
      *-commutative [<=] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{1 \cdot \left(-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      distribute-lft-in [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{1 \cdot -1 + 1 \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{-1} + 1 \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      *-commutative [<=] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + \color{blue}{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot 1}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      *-rgt-identity [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + \color{blue}{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      associate-+r+ [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\color{blue}{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + 1\right) + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}}}}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\color{blue}{\left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}}}{2} \]
      associate-+l+ [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{\color{blue}{1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}}{2} \]
      *-rgt-identity [<=] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \left(\color{blue}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot 1} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)}}{2} \]
      unpow2 [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot 1 + \color{blue}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}\right)}}{2} \]
      distribute-lft-in [<=] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}{1 + \color{blue}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

   6. Applied egg-rr 2.5

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

   7. Simplified 0.0

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

      [Start] 2.5	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + {\alpha}^{3} \cdot \frac{1}{{\left(\alpha + \left(\beta + 2\right)\right)}^{3}}}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      *-commutative [<=] 2.5	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + \color{blue}{\frac{1}{{\left(\alpha + \left(\beta + 2\right)\right)}^{3}} \cdot {\alpha}^{3}}}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      associate-/r/ [<=] 2.5	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + \color{blue}{\frac{1}{\frac{{\left(\alpha + \left(\beta + 2\right)\right)}^{3}}{{\alpha}^{3}}}}}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
      cube-div [<=] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 + \frac{1}{\color{blue}{{\left(\frac{\alpha + \left(\beta + 2\right)}{\alpha}\right)}^{3}}}}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]

   8. Applied egg-rr 0.0

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{\frac{1}{\frac{-1 - {\left(\frac{\alpha + \left(\beta + 2\right)}{\alpha}\right)}^{-3}}{1 - {\left(\frac{\alpha + \left(\beta + 2\right)}{\alpha}\right)}^{-6}}}}}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\frac{-1}{\frac{-1 - {\left(\frac{\alpha + \left(\beta + 2\right)}{\alpha}\right)}^{-3}}{1 - {\left(\frac{\alpha + \left(\beta + 2\right)}{\alpha}\right)}^{-6}}}}{1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	16708

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha}{t_0}\\ t_2 := \frac{t_0}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\frac{1 - {t_2}^{-6}}{1 + {t_2}^{-3}}}{1 + t_1 \cdot \left(1 + t_1\right)}}{2}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	14660

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{-1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}}\right)}{2}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7620

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 6
Error	7.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 95000000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	4.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 95000000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	4.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	30.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	38.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.2 \cdot 10^{+118}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]

Alternative 11
Error	40.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023083 
(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))