Octave 3.8, jcobi/3

Average Error: 3.6 → 0.3

Time: 32.5s

Precision: binary64

Cost: 1732

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

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ 2.0 alpha))))
   (if (<= beta 5.7e+143)
     (/ (* (+ beta 1.0) (/ (+ 1.0 alpha) (+ beta (+ 3.0 alpha)))) (* t_0 t_0))
     (* 0.5 (- (/ (/ -2.0 (+ alpha (+ 3.0 beta))) (/ beta (+ 1.0 alpha))))))))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = beta + (2.0 + alpha);
	double tmp;
	if (beta <= 5.7e+143) {
		tmp = ((beta + 1.0) * ((1.0 + alpha) / (beta + (3.0 + alpha)))) / (t_0 * t_0);
	} else {
		tmp = 0.5 * -((-2.0 / (alpha + (3.0 + beta))) / (beta / (1.0 + alpha)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / ((alpha + beta) + (2.0d0 * 1.0d0))) / ((alpha + beta) + (2.0d0 * 1.0d0))) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (2.0d0 + alpha)
    if (beta <= 5.7d+143) then
        tmp = ((beta + 1.0d0) * ((1.0d0 + alpha) / (beta + (3.0d0 + alpha)))) / (t_0 * t_0)
    else
        tmp = 0.5d0 * -(((-2.0d0) / (alpha + (3.0d0 + beta))) / (beta / (1.0d0 + alpha)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

public static double code(double alpha, double beta) {
	double t_0 = beta + (2.0 + alpha);
	double tmp;
	if (beta <= 5.7e+143) {
		tmp = ((beta + 1.0) * ((1.0 + alpha) / (beta + (3.0 + alpha)))) / (t_0 * t_0);
	} else {
		tmp = 0.5 * -((-2.0 / (alpha + (3.0 + beta))) / (beta / (1.0 + alpha)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)

↓

def code(alpha, beta):
	t_0 = beta + (2.0 + alpha)
	tmp = 0
	if beta <= 5.7e+143:
		tmp = ((beta + 1.0) * ((1.0 + alpha) / (beta + (3.0 + alpha)))) / (t_0 * t_0)
	else:
		tmp = 0.5 * -((-2.0 / (alpha + (3.0 + beta))) / (beta / (1.0 + alpha)))
	return tmp

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(beta + Float64(2.0 + alpha))
	tmp = 0.0
	if (beta <= 5.7e+143)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(Float64(1.0 + alpha) / Float64(beta + Float64(3.0 + alpha)))) / Float64(t_0 * t_0));
	else
		tmp = Float64(0.5 * Float64(-Float64(Float64(-2.0 / Float64(alpha + Float64(3.0 + beta))) / Float64(beta / Float64(1.0 + alpha)))));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
end

↓

function tmp_2 = code(alpha, beta)
	t_0 = beta + (2.0 + alpha);
	tmp = 0.0;
	if (beta <= 5.7e+143)
		tmp = ((beta + 1.0) * ((1.0 + alpha) / (beta + (3.0 + alpha)))) / (t_0 * t_0);
	else
		tmp = 0.5 * -((-2.0 / (alpha + (3.0 + beta))) / (beta / (1.0 + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.7e+143], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * (-N[(N[(-2.0 / N[(alpha + N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.70000000000000022e143

   1. Initial program 0.2

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

   2. Simplified 0.1

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

      [Start] 0.2	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-47 [=>] 0.2	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-44 [=>] 0.1	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 6.6

      \[\leadsto \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} + 0} \]

   4. Simplified 0.2

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

      [Start] 6.6	\[ \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} + 0 \]
      rational.json-simplify-4 [=>] 6.6	\[ \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
      rational.json-simplify-46 [=>] 0.1	\[ \color{blue}{\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-2 [=>] 0.1	\[ \frac{\frac{\color{blue}{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-49 [=>] 0.2	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-41 [=>] 0.2	\[ \frac{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\color{blue}{\beta + \left(3 + \alpha\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-41 [=>] 0.2	\[ \frac{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{\color{blue}{\left(\beta + \left(2 + \alpha\right)\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-41 [=>] 0.2	\[ \frac{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \color{blue}{\left(\beta + \left(2 + \alpha\right)\right)}} \]

   if 5.70000000000000022e143 < beta

   1. Initial program 10.7

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

   2. Simplified 13.2

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

      [Start] 10.7	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-47 [=>] 13.2	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-44 [=>] 13.2	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 63.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}{\left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}\right)}} \]

   4. Simplified 1.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{2}{\left(\beta + \left(3 + \alpha\right)\right) \cdot \left(\left(\beta + \left(2 + \alpha\right)\right) \cdot \frac{\frac{\beta + \left(2 + \alpha\right)}{\beta + 1}}{1 + \alpha}\right)}} \]
      Proof

      [Start] 63.6

      \[ 0.5 \cdot \frac{2 \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}{\left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}\right)} \]

      rational.json-simplify-46 [=>] 60.6

      \[ 0.5 \cdot \color{blue}{\frac{\frac{2 \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}} \]

   5. Applied egg-rr 0.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\frac{2}{\beta + \left(3 + \alpha\right)}}{-\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right)} \]

   6. Simplified 0.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \frac{\frac{2 + \left(\alpha + \beta\right)}{1 + \alpha}}{1 + \beta}}\right)} \]
      Proof

      [Start] 0.1	\[ 0.5 \cdot \left(-1 \cdot \frac{\frac{2}{\beta + \left(3 + \alpha\right)}}{-\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right) \]
      rational.json-simplify-2 [=>] 0.1	\[ 0.5 \cdot \color{blue}{\left(\frac{\frac{2}{\beta + \left(3 + \alpha\right)}}{-\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 0.1	\[ 0.5 \cdot \color{blue}{\left(-\frac{\frac{2}{\beta + \left(3 + \alpha\right)}}{-\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right)} \]
      rational.json-simplify-8 [=>] 0.1	\[ 0.5 \cdot \left(-\frac{\frac{2}{\beta + \left(3 + \alpha\right)}}{\color{blue}{\left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}\right) \cdot -1}}\right) \]
      rational.json-simplify-2 [=>] 0.1	\[ 0.5 \cdot \left(-\frac{\frac{2}{\beta + \left(3 + \alpha\right)}}{\color{blue}{-1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}\right)}}\right) \]
      rational.json-simplify-46 [=>] 0.1	\[ 0.5 \cdot \left(-\color{blue}{\frac{\frac{\frac{2}{\beta + \left(3 + \alpha\right)}}{-1}}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}}\right) \]
      rational.json-simplify-44 [=>] 0.1	\[ 0.5 \cdot \left(-\frac{\color{blue}{\frac{\frac{2}{-1}}{\beta + \left(3 + \alpha\right)}}}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right) \]
      metadata-eval [=>] 0.1	\[ 0.5 \cdot \left(-\frac{\frac{\color{blue}{-2}}{\beta + \left(3 + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right) \]
      rational.json-simplify-1 [=>] 0.1	\[ 0.5 \cdot \left(-\frac{\frac{-2}{\beta + \color{blue}{\left(\alpha + 3\right)}}}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right) \]
      rational.json-simplify-41 [=>] 0.1	\[ 0.5 \cdot \left(-\frac{\frac{-2}{\color{blue}{\alpha + \left(3 + \beta\right)}}}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right) \]
      rational.json-simplify-1 [=>] 0.1	\[ 0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\left(2 + \color{blue}{\left(\alpha + \beta\right)}\right) \cdot \frac{\frac{2 + \left(\beta + \alpha\right)}{\alpha + 1}}{\beta + 1}}\right) \]

   7. Taylor expanded in beta around inf 0.5

      \[\leadsto 0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\color{blue}{\frac{\beta}{1 + \alpha}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1920

\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ 0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{t_0 \cdot \frac{\frac{t_0}{1 + \alpha}}{1 + \beta}}\right) \end{array} \]

Alternative 2
Error	0.5
Cost	1856

\[\begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ 0.5 \cdot \frac{2}{\left(\beta + \left(3 + \alpha\right)\right) \cdot \left(t_0 \cdot \frac{\frac{t_0}{\beta + 1}}{1 + \alpha}\right)} \end{array} \]

Alternative 3
Error	0.4
Cost	1732

\[\begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	1604

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\beta + 3} \cdot 0.5}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

Alternative 5
Error	1.1
Cost	1476

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3} \cdot \left(\beta + 1\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

Alternative 6
Error	1.0
Cost	1348

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 3}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

Alternative 7
Error	1.9
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.2222222222222222 \cdot \beta + 0.3333333333333333}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

Alternative 8
Error	1.9
Cost	1156

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-\frac{\frac{-2}{\alpha + \left(3 + \beta\right)}}{\frac{\beta}{1 + \alpha}}\right)\\ \end{array} \]

Alternative 9
Error	1.9
Cost	900

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.82:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{-1 - \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 10
Error	5.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.85:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{2}{3 + \beta}}{2 + \beta}\\ \end{array} \]

Alternative 11
Error	3.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\beta + \left(3 + \alpha\right)\right)}\\ \end{array} \]

Alternative 12
Error	1.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 13
Error	5.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 14
Error	5.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.85:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 15
Error	34.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2222222222222222}{\beta}\\ \end{array} \]

Alternative 16
Error	32.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{-51}:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\alpha}}{\beta}\\ \end{array} \]

Alternative 17
Error	34.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2222222222222222}{\beta}\\ \end{array} \]

Alternative 18
Error	35.5
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023068 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))