Octave 3.8, jcobi/3

Average Error: 3.3 → 0.2

Time: 39.3s

Precision: binary64

Cost: 3780

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t_0}}{t_0}}{t_0 + 1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{t_0}}{t_0}}{\left(\beta + 1\right) - \left(-2 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

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 (+ (+ alpha beta) 2.0)))
   (if (<=
        (/
         (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0)
         (+ t_0 1.0))
        0.1)
     (/
      (/ (/ (+ (+ (+ alpha beta) (* alpha beta)) 1.0) t_0) t_0)
      (- (+ beta 1.0) (- -2.0 alpha)))
     (/ (/ (- alpha -1.0) (+ alpha (+ beta 2.0))) (+ (+ alpha beta) 3.0)))))

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 = (alpha + beta) + 2.0;
	double tmp;
	if (((((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)) <= 0.1) {
		tmp = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / ((beta + 1.0) - (-2.0 - alpha));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / ((alpha + beta) + 3.0);
	}
	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 = (alpha + beta) + 2.0d0
    if (((((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)) <= 0.1d0) then
        tmp = (((((alpha + beta) + (alpha * beta)) + 1.0d0) / t_0) / t_0) / ((beta + 1.0d0) - ((-2.0d0) - alpha))
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (beta + 2.0d0))) / ((alpha + beta) + 3.0d0)
    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 = (alpha + beta) + 2.0;
	double tmp;
	if (((((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)) <= 0.1) {
		tmp = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / ((beta + 1.0) - (-2.0 - alpha));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / ((alpha + beta) + 3.0);
	}
	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 = (alpha + beta) + 2.0
	tmp = 0
	if ((((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)) <= 0.1:
		tmp = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / ((beta + 1.0) - (-2.0 - alpha))
	else:
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / ((alpha + beta) + 3.0)
	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(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0)) <= 0.1)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(alpha * beta)) + 1.0) / t_0) / t_0) / Float64(Float64(beta + 1.0) - Float64(-2.0 - alpha)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(alpha + beta) + 3.0));
	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 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (((((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)) <= 0.1)
		tmp = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / ((beta + 1.0) - (-2.0 - alpha));
	else
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / ((alpha + beta) + 3.0);
	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[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(beta + 1.0), $MachinePrecision] - N[(-2.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) < 0.10000000000000001

   1. Initial program 0.1

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

   2. Simplified 0.1

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

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

   3. Applied egg-rr 0.1

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

   if 0.10000000000000001 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1))

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-1 [=>] 64.0	\[ \frac{\frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-41 [=>] 64.0	\[ \frac{\frac{\frac{\color{blue}{\left(\alpha + \beta\right) + \left(\beta \cdot \alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\color{blue}{\alpha \cdot \beta} + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      metadata-eval [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-1 [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-41 [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      metadata-eval [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-1 [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-41 [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-1 [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      metadata-eval [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{1 + \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]
      rational.json-simplify-41 [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]
      metadata-eval [=>] 64.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

   3. Taylor expanded in beta around inf 2.6

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

   4. Simplified 2.6

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

      [Start] 2.6	\[ \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 3} \]
      rational.json-simplify-17 [=>] 2.6	\[ \frac{\frac{\color{blue}{\alpha - -1}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 3} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\beta + 1\right) - \left(-2 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1988

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+103}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{t_0}}{t_0}}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1732

\[\begin{array}{l} t_0 := \beta - \left(-2 - \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+103}:\\ \;\;\;\;\frac{\frac{\frac{\left(-1 - \beta\right) \cdot \left(-1 - \alpha\right)}{t_0}}{t_0}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\beta + 1\right) - \left(-2 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternative 4
Error	1.0
Cost	1348

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{t_0}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_1}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	1092

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{t_0}\\ \end{array} \]

Alternative 6
Error	1.7
Cost	964

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 11.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 8
Error	1.7
Cost	836

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{t_0}\\ \end{array} \]

Alternative 9
Error	2.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Error	28.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.5:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	28.7
Cost	448

\[\frac{\frac{\alpha - -1}{\beta}}{\beta} \]

Alternative 12
Error	42.1
Cost	320

\[\frac{\frac{\alpha}{\beta}}{\beta} \]

Alternative 13
Error	62.4
Cost	192

\[\frac{1}{\alpha} \]

Alternative 14
Error	60.2
Cost	192

\[\frac{1}{\beta} \]

Reproduce

herbie shell --seed 2023077 
(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)))