Octave 3.8, jcobi/1

Average Accuracy: 74.2% → 98.5%

Time: 17.5s

Precision: binary64

Cost: 28676

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta + 2}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9996:\\ \;\;\;\;\frac{{t_0}^{3} + \left(\frac{\beta}{{\alpha}^{3}} \cdot {\left(\beta + 2\right)}^{2} + \mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, t_0, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ beta 2.0) alpha)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9996)
     (/
      (+
       (pow t_0 3.0)
       (+
        (* (/ beta (pow alpha 3.0)) (pow (+ beta 2.0) 2.0))
        (fma
         (/ (- (- -2.0 beta) beta) alpha)
         t_0
         (/ (+ beta (+ beta 2.0)) alpha))))
      2.0)
     (/
      (-
       (/ beta (+ beta (+ alpha 2.0)))
       (log (exp (+ (/ alpha (+ alpha (+ beta 2.0))) -1.0))))
      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 + 2.0) / alpha;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9996) {
		tmp = (pow(t_0, 3.0) + (((beta / pow(alpha, 3.0)) * pow((beta + 2.0), 2.0)) + fma((((-2.0 - beta) - beta) / alpha), t_0, ((beta + (beta + 2.0)) / alpha)))) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) - log(exp(((alpha / (alpha + (beta + 2.0))) + -1.0)))) / 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(Float64(beta + 2.0) / alpha)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9996)
		tmp = Float64(Float64((t_0 ^ 3.0) + Float64(Float64(Float64(beta / (alpha ^ 3.0)) * (Float64(beta + 2.0) ^ 2.0)) + fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), t_0, Float64(Float64(beta + Float64(beta + 2.0)) / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) - log(exp(Float64(Float64(alpha / Float64(alpha + Float64(beta + 2.0))) + -1.0)))) / 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[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9996], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[(N[(N[(beta / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(beta + 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * t$95$0 + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[Exp[N[(N[(alpha / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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)) < -0.99960000000000004

   1. Initial program 8.4%

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

   2. Simplified 8.4%

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

      [Start] 8.4	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 8.4	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 98.5%

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

   4. Simplified 100.0%

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

      [Start] 98.5	\[ \frac{\frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      cube-mult [=>] 98.5	\[ \frac{\frac{\color{blue}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}}{{\alpha}^{3}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      unpow2 [<=] 98.5	\[ \frac{\frac{\left(\beta + 2\right) \cdot \color{blue}{{\left(\beta + 2\right)}^{2}}}{{\alpha}^{3}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      cube-mult [=>] 98.5	\[ \frac{\frac{\left(\beta + 2\right) \cdot {\left(\beta + 2\right)}^{2}}{\color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      unpow2 [<=] 98.5	\[ \frac{\frac{\left(\beta + 2\right) \cdot {\left(\beta + 2\right)}^{2}}{\alpha \cdot \color{blue}{{\alpha}^{2}}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      times-frac [=>] 98.5	\[ \frac{\color{blue}{\frac{\beta + 2}{\alpha} \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      unpow2 [=>] 98.5	\[ \frac{\frac{\beta + 2}{\alpha} \cdot \frac{\color{blue}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{{\alpha}^{2}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      unpow2 [=>] 98.5	\[ \frac{\frac{\beta + 2}{\alpha} \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}{\color{blue}{\alpha \cdot \alpha}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      times-frac [=>] 98.5	\[ \frac{\frac{\beta + 2}{\alpha} \cdot \color{blue}{\left(\frac{\beta + 2}{\alpha} \cdot \frac{\beta + 2}{\alpha}\right)} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      cube-unmult [=>] 98.5	\[ \frac{\color{blue}{{\left(\frac{\beta + 2}{\alpha}\right)}^{3}} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      +-commutative [=>] 98.5	\[ \frac{{\left(\frac{\color{blue}{2 + \beta}}{\alpha}\right)}^{3} + \left(-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right)\right)}{2} \]
      +-commutative [=>] 98.5	\[ \frac{{\left(\frac{2 + \beta}{\alpha}\right)}^{3} + \color{blue}{\left(\left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta \cdot {\left(\beta + 2\right)}^{2}}{{\alpha}^{3}}\right) + -1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}\right)}}{2} \]

   if -0.99960000000000004 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 99.8	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 99.8%

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

      [Start] 99.8	\[ \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2} \]
      div-sub [=>] 99.8	\[ \frac{\color{blue}{\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2} \]
      associate-+l- [=>] 99.8	\[ \frac{\color{blue}{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2} \]
      associate-+l+ [=>] 99.8	\[ \frac{\frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2} \]
      associate-+l+ [=>] 99.8	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} - 1\right)}{2} \]

   4. Applied egg-rr 99.8%

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

      [Start] 99.8	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}{2} \]
      add-log-exp [=>] 99.8	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\log \left(e^{\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1}\right)}}{2} \]
      sub-neg [=>] 99.8	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\color{blue}{\frac{\alpha}{\beta + \left(\alpha + 2\right)} + \left(-1\right)}}\right)}{2} \]
      +-commutative [=>] 99.8	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\color{blue}{\left(\alpha + 2\right) + \beta}} + \left(-1\right)}\right)}{2} \]
      associate-+l+ [=>] 99.8	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\color{blue}{\alpha + \left(2 + \beta\right)}} + \left(-1\right)}\right)}{2} \]
      metadata-eval [=>] 99.8	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\alpha + \left(2 + \beta\right)} + \color{blue}{-1}}\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9996:\\ \;\;\;\;\frac{{\left(\frac{\beta + 2}{\alpha}\right)}^{3} + \left(\frac{\beta}{{\alpha}^{3}} \cdot {\left(\beta + 2\right)}^{2} + \mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.5%
Cost	14660

\[\begin{array}{l} t_0 := \frac{\beta}{\beta + \left(\alpha + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9996:\\ \;\;\;\;\frac{t_0 + \left(\frac{\beta + 2}{\alpha} - \frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}\right)}{2}\\ \end{array} \]

Alternative 2
Accuracy	98.5%
Cost	8580

\[\begin{array}{l} t_0 := \frac{\beta}{\beta + \left(\alpha + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9996:\\ \;\;\;\;\frac{t_0 + \left(\frac{\beta + 2}{\alpha} - \frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - \mathsf{fma}\left(\frac{1}{\alpha + \left(\beta + 2\right)}, \alpha, -1\right)}{2}\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	8388

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9996:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \mathsf{fma}\left(\frac{1}{\alpha + \left(\beta + 2\right)}, \alpha, -1\right)}{2}\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	1988

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\alpha + \left(\beta + 2\right)} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{t_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 6
Accuracy	99.7%
Cost	1604

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 7
Accuracy	99.7%
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 8
Accuracy	83.5%
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 65 \lor \neg \left(\alpha \leq 2.8 \cdot 10^{+32}\right) \land \alpha \leq 3.6 \cdot 10^{+113}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Accuracy	92.5%
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 65:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 5.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Accuracy	92.7%
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 47:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{2 - \frac{4}{\alpha}}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 5.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Accuracy	67.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -5.4 \cdot 10^{-241}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	70.9%
Cost	196

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

Alternative 13
Accuracy	49.6%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023160 
(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))