Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.8%

Time: 25.3s

Precision: binary64

Cost: 28612

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta - \alpha}{t_0}\\ t_2 := \frac{\beta + \left(\beta + 2\right)}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{t_2 - \frac{\beta + 2}{\alpha} \cdot t_2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {t_1}^{3}}\right)}{\left(1 + {t_1}^{2}\right) + \frac{\alpha - \beta}{t_0}}}{2}\\ \end{array} \]

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) t_0))
        (t_2 (/ (+ beta (+ beta 2.0)) alpha)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
     (/ (- t_2 (* (/ (+ beta 2.0) alpha) t_2)) 2.0)
     (/
      (/
       (log (exp (+ 1.0 (pow t_1 3.0))))
       (+ (+ 1.0 (pow t_1 2.0)) (/ (- alpha beta) t_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) / t_0;
	double t_2 = (beta + (beta + 2.0)) / alpha;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = (t_2 - (((beta + 2.0) / alpha) * t_2)) / 2.0;
	} else {
		tmp = (log(exp((1.0 + pow(t_1, 3.0)))) / ((1.0 + pow(t_1, 2.0)) + ((alpha - beta) / t_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) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = (beta - alpha) / t_0
    t_2 = (beta + (beta + 2.0d0)) / alpha
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999d0)) then
        tmp = (t_2 - (((beta + 2.0d0) / alpha) * t_2)) / 2.0d0
    else
        tmp = (log(exp((1.0d0 + (t_1 ** 3.0d0)))) / ((1.0d0 + (t_1 ** 2.0d0)) + ((alpha - beta) / t_0))) / 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) / t_0;
	double t_2 = (beta + (beta + 2.0)) / alpha;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = (t_2 - (((beta + 2.0) / alpha) * t_2)) / 2.0;
	} else {
		tmp = (Math.log(Math.exp((1.0 + Math.pow(t_1, 3.0)))) / ((1.0 + Math.pow(t_1, 2.0)) + ((alpha - beta) / t_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) / t_0
	t_2 = (beta + (beta + 2.0)) / alpha
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999:
		tmp = (t_2 - (((beta + 2.0) / alpha) * t_2)) / 2.0
	else:
		tmp = (math.log(math.exp((1.0 + math.pow(t_1, 3.0)))) / ((1.0 + math.pow(t_1, 2.0)) + ((alpha - beta) / t_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) / t_0)
	t_2 = Float64(Float64(beta + Float64(beta + 2.0)) / alpha)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999)
		tmp = Float64(Float64(t_2 - Float64(Float64(Float64(beta + 2.0) / alpha) * t_2)) / 2.0);
	else
		tmp = Float64(Float64(log(exp(Float64(1.0 + (t_1 ^ 3.0)))) / Float64(Float64(1.0 + (t_1 ^ 2.0)) + Float64(Float64(alpha - beta) / t_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) / t_0;
	t_2 = (beta + (beta + 2.0)) / alpha;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999)
		tmp = (t_2 - (((beta + 2.0) / alpha) * t_2)) / 2.0;
	else
		tmp = (log(exp((1.0 + (t_1 ^ 3.0)))) / ((1.0 + (t_1 ^ 2.0)) + ((alpha - beta) / t_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] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(N[(t$95$2 - N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Log[N[Exp[N[(1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.998999999999999999

   1. Initial program 8.8%

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

   2. Simplified 8.8%

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

      [Start] 8.8%	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 8.8%	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 91.6%

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

   4. Simplified 99.4%

      \[\leadsto \frac{\color{blue}{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\beta + 2}{\alpha} \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha}}}{2} \]
      Step-by-step derivation

      [Start] 91.6%	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      mul-1-neg [=>] 91.6%	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \color{blue}{\left(-\frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      unsub-neg [=>] 91.6%	\[ \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} - \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}}{2} \]
      associate-*r/ [=>] 91.6%	\[ \frac{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \beta - \left(\beta + 2\right)\right)}{\alpha}} - \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      mul-1-neg [=>] 91.6%	\[ \frac{\frac{\color{blue}{-\left(-1 \cdot \beta - \left(\beta + 2\right)\right)}}{\alpha} - \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      mul-1-neg [=>] 91.6%	\[ \frac{\frac{-\left(\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)\right)}{\alpha} - \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      +-commutative [=>] 91.6%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\color{blue}{\beta \cdot \left(\beta + 2\right) + {\left(\beta + 2\right)}^{2}}}{{\alpha}^{2}}}{2} \]
      unpow2 [=>] 91.6%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\beta \cdot \left(\beta + 2\right) + \color{blue}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{{\alpha}^{2}}}{2} \]
      distribute-rgt-in [<=] 91.6%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\color{blue}{\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)}}{{\alpha}^{2}}}{2} \]
      *-lft-identity [<=] 91.6%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\left(\beta + 2\right) \cdot \left(\beta + \color{blue}{1 \cdot \left(\beta + 2\right)}\right)}{{\alpha}^{2}}}{2} \]
      metadata-eval [<=] 91.6%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\left(\beta + 2\right) \cdot \left(\beta + \color{blue}{\left(--1\right)} \cdot \left(\beta + 2\right)\right)}{{\alpha}^{2}}}{2} \]
      cancel-sign-sub-inv [<=] 91.6%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\left(\beta + 2\right) \cdot \color{blue}{\left(\beta - -1 \cdot \left(\beta + 2\right)\right)}}{{\alpha}^{2}}}{2} \]
      unpow2 [=>] 91.6%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \frac{\left(\beta + 2\right) \cdot \left(\beta - -1 \cdot \left(\beta + 2\right)\right)}{\color{blue}{\alpha \cdot \alpha}}}{2} \]
      times-frac [=>] 99.4%	\[ \frac{\frac{-\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}{\alpha} - \color{blue}{\frac{\beta + 2}{\alpha} \cdot \frac{\beta - -1 \cdot \left(\beta + 2\right)}{\alpha}}}{2} \]

   if -0.998999999999999999 < (/.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%

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

      [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%

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

      [Start] 99.9%	\[ \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2} \]
      flip3-+ [=>] 99.9%	\[ \frac{\color{blue}{\frac{{\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)}}}{2} \]
      pow3 [<=] 99.9%	\[ \frac{\frac{\color{blue}{\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}} + {1}^{3}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)}}{2} \]
      metadata-eval [=>] 99.9%	\[ \frac{\frac{\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \color{blue}{1}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)}}{2} \]
      +-commutative [=>] 99.9%	\[ \frac{\frac{\color{blue}{1 + \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)}}{2} \]
      pow3 [=>] 99.9%	\[ \frac{\frac{1 + \color{blue}{{\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)}}{2} \]
      associate-+l+ [=>] 99.9%	\[ \frac{\frac{1 + {\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)}^{3}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)}}{2} \]
      metadata-eval [=>] 99.9%	\[ \frac{\frac{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(\color{blue}{1} - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)}}{2} \]
      *-rgt-identity [=>] 99.9%	\[ \frac{\frac{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + \left(1 - \color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)}}{2} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}\right)}}{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + 1\right) - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
      Step-by-step derivation

      [Start] 99.9%	\[ \frac{\frac{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + 1\right) - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
      add-log-exp [=>] 99.9%	\[ \frac{\frac{\color{blue}{\log \left(e^{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}\right)}}{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + 1\right) - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} - \frac{\beta + 2}{\alpha} \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}\right)}{\left(1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}\right) + \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	28612

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta - \alpha}{t_0}\\ t_2 := \frac{\beta + \left(\beta + 2\right)}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{t_2 - \frac{\beta + 2}{\alpha} \cdot t_2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {t_1}^{3}}\right)}{\left(1 + {t_1}^{2}\right) + \frac{\alpha - \beta}{t_0}}}{2}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	2116

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(\frac{\beta + 2}{\alpha} - \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{t_0}{\beta - \alpha}}}{2}\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	2116

\[\begin{array}{l} t_0 := \frac{\beta + \left(\beta + 2\right)}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{t_0 - \frac{\beta + 2}{\alpha} \cdot t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\beta + \left(\alpha + 2\right)}{\beta - \alpha}}}{2}\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{t_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(t_1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 5
Accuracy	99.6%
Cost	1604

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

Alternative 6
Accuracy	99.6%
Cost	1604

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

Alternative 7
Accuracy	99.6%
Cost	1476

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

Alternative 8
Accuracy	70.8%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -2.65 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.08 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	87.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 980:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.7 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\alpha} - \frac{2}{\alpha \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Accuracy	92.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 11
Accuracy	92.3%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 12
Accuracy	92.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 13
Accuracy	70.3%
Cost	460

\[\begin{array}{l} \mathbf{if}\;\beta \leq -9.5 \cdot 10^{-117}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -6.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	71.3%
Cost	196

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

Alternative 15
Accuracy	49.1%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023269 
(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))