Octave 3.8, jcobi/3

Average Error: 5.74% → 0.18%

Time: 24.1s

Precision: binary64

Cost: 1732

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)}}{\frac{t_0}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{1}{\beta} + \left(1 + \frac{\alpha}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

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 (+ alpha 2.0))))
   (if (<= beta 2e+130)
     (/ (/ (+ alpha 1.0) (* t_0 (+ beta (+ alpha 3.0)))) (/ t_0 (+ 1.0 beta)))
     (/
      (/
       (+ alpha 1.0)
       (* (+ alpha (+ beta 2.0)) (+ (/ 1.0 beta) (+ 1.0 (/ alpha beta)))))
      (+ 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 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 2e+130) {
		tmp = ((alpha + 1.0) / (t_0 * (beta + (alpha + 3.0)))) / (t_0 / (1.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + (beta + 2.0)) * ((1.0 / beta) + (1.0 + (alpha / beta))))) / (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 = beta + (alpha + 2.0d0)
    if (beta <= 2d+130) then
        tmp = ((alpha + 1.0d0) / (t_0 * (beta + (alpha + 3.0d0)))) / (t_0 / (1.0d0 + beta))
    else
        tmp = ((alpha + 1.0d0) / ((alpha + (beta + 2.0d0)) * ((1.0d0 / beta) + (1.0d0 + (alpha / beta))))) / (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 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 2e+130) {
		tmp = ((alpha + 1.0) / (t_0 * (beta + (alpha + 3.0)))) / (t_0 / (1.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + (beta + 2.0)) * ((1.0 / beta) + (1.0 + (alpha / beta))))) / (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 = beta + (alpha + 2.0)
	tmp = 0
	if beta <= 2e+130:
		tmp = ((alpha + 1.0) / (t_0 * (beta + (alpha + 3.0)))) / (t_0 / (1.0 + beta))
	else:
		tmp = ((alpha + 1.0) / ((alpha + (beta + 2.0)) * ((1.0 / beta) + (1.0 + (alpha / beta))))) / (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(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (beta <= 2e+130)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(t_0 * Float64(beta + Float64(alpha + 3.0)))) / Float64(t_0 / Float64(1.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(1.0 / beta) + Float64(1.0 + Float64(alpha / beta))))) / Float64(alpha + Float64(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 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (beta <= 2e+130)
		tmp = ((alpha + 1.0) / (t_0 * (beta + (alpha + 3.0)))) / (t_0 / (1.0 + beta));
	else
		tmp = ((alpha + 1.0) / ((alpha + (beta + 2.0)) * ((1.0 / beta) + (1.0 + (alpha / beta))))) / (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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+130], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / beta), $MachinePrecision] + N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000001e130

   1. Initial program 0.23

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

   2. Simplified 0.22

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

      [Start] 0.23	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 0.27	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/r* [<=] 7.29	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
      associate-/l/ [<=] 0.23	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 0.32

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

   4. Simplified 0.34

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

      [Start] 0.32	\[ \frac{\beta + 1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \frac{\beta + \left(\alpha + 3\right)}{1 + \alpha}} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      *-commutative [=>] 0.32	\[ \frac{\beta + 1}{\color{blue}{\frac{\beta + \left(\alpha + 3\right)}{1 + \alpha} \cdot \left(\beta + \left(\alpha + 2\right)\right)}} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      associate-/r* [=>] 0.34	\[ \color{blue}{\frac{\frac{\beta + 1}{\frac{\beta + \left(\alpha + 3\right)}{1 + \alpha}}}{\beta + \left(\alpha + 2\right)}} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      +-commutative [=>] 0.34	\[ \frac{\frac{\color{blue}{1 + \beta}}{\frac{\beta + \left(\alpha + 3\right)}{1 + \alpha}}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      +-commutative [=>] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\color{blue}{\left(\alpha + 3\right) + \beta}}{1 + \alpha}}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      associate-+l+ [=>] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\color{blue}{\alpha + \left(3 + \beta\right)}}{1 + \alpha}}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      associate-+r+ [=>] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\alpha + \left(3 + \beta\right)}{1 + \alpha}}}{\color{blue}{\left(\beta + \alpha\right) + 2}} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      +-commutative [=>] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\alpha + \left(3 + \beta\right)}{1 + \alpha}}}{\color{blue}{\left(\alpha + \beta\right)} + 2} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      associate-+r+ [<=] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\alpha + \left(3 + \beta\right)}{1 + \alpha}}}{\color{blue}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{\beta + \left(\alpha + 2\right)} \]
      associate-+r+ [=>] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\alpha + \left(3 + \beta\right)}{1 + \alpha}}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\left(\beta + \alpha\right) + 2}} \]
      +-commutative [=>] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\alpha + \left(3 + \beta\right)}{1 + \alpha}}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2} \]
      associate-+r+ [<=] 0.34	\[ \frac{\frac{1 + \beta}{\frac{\alpha + \left(3 + \beta\right)}{1 + \alpha}}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\alpha + \left(\beta + 2\right)}} \]

   5. Applied egg-rr 30.45

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)}}{\frac{\beta + \left(\alpha + 2\right)}{1 + \alpha}}\right)} - 1\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]

   6. Simplified 0.32

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

      [Start] 30.45	\[ \left(e^{\mathsf{log1p}\left(\frac{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)}}{\frac{\beta + \left(\alpha + 2\right)}{1 + \alpha}}\right)} - 1\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      expm1-def [=>] 0.34	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)}}{\frac{\beta + \left(\alpha + 2\right)}{1 + \alpha}}\right)\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      expm1-log1p [=>] 0.33	\[ \color{blue}{\frac{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)}}{\frac{\beta + \left(\alpha + 2\right)}{1 + \alpha}}} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      associate-/r/ [=>] 0.34	\[ \color{blue}{\left(\frac{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)}}{\beta + \left(\alpha + 2\right)} \cdot \left(1 + \alpha\right)\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      associate-*l/ [=>] 0.33	\[ \color{blue}{\frac{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)} \cdot \left(1 + \alpha\right)}{\beta + \left(\alpha + 2\right)}} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      associate-*l/ [=>] 0.34	\[ \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\beta + \left(\alpha + 3\right)}}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      associate-*r/ [<=] 0.34	\[ \frac{\color{blue}{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      associate-*r/ [<=] 0.34	\[ \color{blue}{\left(\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{\beta + \left(\alpha + 2\right)}\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      associate-/l/ [=>] 0.32	\[ \left(\left(1 + \beta\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      associate-+r+ [=>] 0.32	\[ \left(\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\color{blue}{\left(\left(\beta + \alpha\right) + 2\right)} \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]
      +-commutative [=>] 0.32	\[ \left(\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \color{blue}{\left(\left(\alpha + 3\right) + \beta\right)}}\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)} \]

   7. Applied egg-rr 0.23

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

   if 2.0000000000000001e130 < beta

   1. Initial program 16.49

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

   2. Simplified 0.08

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

      [Start] 16.49	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Taylor expanded in beta around inf 0.08

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

4. Final simplification 0.18

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}{\frac{\beta + \left(\alpha + 2\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{1}{\beta} + \left(1 + \frac{\alpha}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.22%
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{3 + \left(\alpha + \beta\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 2
Error	0.22%
Cost	1732

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)}}{\frac{t_0}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3
Error	0.18%
Cost	1600

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

Alternative 4
Error	1.06%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 65000000:\\ \;\;\;\;\left(\left(1 + \beta\right) \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 5
Error	1.09%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 26000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6
Error	1.36%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	2.8%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.56:\\ \;\;\;\;\frac{1}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right) \cdot \left(2 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8
Error	3.23%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.1:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9
Error	3.27%
Cost	708

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

Alternative 10
Error	42.66%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{0.3333333333333333}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	40.17%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.1:\\ \;\;\;\;\frac{0.3333333333333333}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12
Error	37.04%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.1:\\ \;\;\;\;\frac{0.3333333333333333}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	84.58%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{0.3333333333333333}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \beta}\\ \end{array} \]

Alternative 14
Error	43.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Error	43.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{\alpha + 1}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Error	42.66%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{\alpha + 1}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	88.31%
Cost	320

\[\frac{0.3333333333333333}{\alpha + 3} \]

Alternative 18
Error	88.49%
Cost	64

\[0.1111111111111111 \]

Reproduce

herbie shell --seed 2023104 
(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)))