Octave 3.8, jcobi/3

Average Error: 3.5 → 0.4

Time: 14.5s

Precision: binary64

Cost: 2500

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\\ \mathbf{if}\;\beta \leq 1.0725411168713128 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \alpha\right) \cdot t_0 + 2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

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 (* (+ alpha (+ beta 2.0)) (+ beta (+ alpha 3.0)))))
   (if (<= beta 1.0725411168713128e+49)
     (/ (* (+ alpha 1.0) (+ beta 1.0)) (+ (* (+ beta alpha) t_0) (* 2.0 t_0)))
     (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ (+ beta alpha) 2.0))))))

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)) * (beta + (alpha + 3.0));
	double tmp;
	if (beta <= 1.0725411168713128e+49) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (((beta + alpha) * t_0) + (2.0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((beta + alpha) + 2.0));
	}
	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)) * (beta + (alpha + 3.0d0))
    if (beta <= 1.0725411168713128d+49) then
        tmp = ((alpha + 1.0d0) * (beta + 1.0d0)) / (((beta + alpha) * t_0) + (2.0d0 * t_0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (1.0d0 + ((beta + alpha) + 2.0d0))
    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)) * (beta + (alpha + 3.0));
	double tmp;
	if (beta <= 1.0725411168713128e+49) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (((beta + alpha) * t_0) + (2.0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((beta + alpha) + 2.0));
	}
	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)) * (beta + (alpha + 3.0))
	tmp = 0
	if beta <= 1.0725411168713128e+49:
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (((beta + alpha) * t_0) + (2.0 * t_0))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((beta + alpha) + 2.0))
	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 + Float64(beta + 2.0)) * Float64(beta + Float64(alpha + 3.0)))
	tmp = 0.0
	if (beta <= 1.0725411168713128e+49)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / Float64(Float64(Float64(beta + alpha) * t_0) + Float64(2.0 * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(1.0 + Float64(Float64(beta + alpha) + 2.0)));
	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)) * (beta + (alpha + 3.0));
	tmp = 0.0;
	if (beta <= 1.0725411168713128e+49)
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (((beta + alpha) * t_0) + (2.0 * t_0));
	else
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((beta + alpha) + 2.0));
	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 + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.0725411168713128e+49], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.07254111687131278e49

   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

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
      Proof
      (/.f64 (*.f64 (+.f64 alpha 1) (+.f64 beta 1)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 alpha)) (+.f64 beta 1)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta alpha) 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta alpha) 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (+.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) 2) 1))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (/.f64 (+.f64 beta 1) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))))): 15 points increase in error, 46 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 10 points increase in error, 16 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 beta 1) (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 beta 1) (+.f64 1 alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 14 points increase in error, 7 points decrease in error
      (/.f64 (/.f64 (*.f64 (+.f64 beta 1) (Rewrite=> +-commutative_binary64 (+.f64 alpha 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (*.f64 (+.f64 beta 1) 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (Rewrite=> *-rgt-identity_binary64 (+.f64 beta 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 beta alpha) alpha)) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 beta alpha) alpha) beta) 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 beta alpha) (+.f64 alpha beta))) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha))) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.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))): 16 points increase in error, 18 points decrease in error

   3. Applied egg-rr 0.1

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

   if 1.07254111687131278e49 < beta

   1. Initial program 7.1

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

   2. Taylor expanded in beta around inf 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.0725411168713128 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \alpha\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right) + 2 \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.4725359173985118 \cdot 10^{+146}:\\ \;\;\;\;\frac{\alpha + 1}{t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\beta + 1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1732

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 1.0725411168713128 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{t_0 \cdot \left(t_0 \cdot \left(\left(\beta + \alpha\right) + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4441594.753858191:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	1348

\[\begin{array}{l} t_0 := 1 + \left(\left(\beta + \alpha\right) + 2\right)\\ \mathbf{if}\;\beta \leq 4441594.753858191:\\ \;\;\;\;\frac{\frac{\beta + 1}{4 + \beta \cdot \left(\beta + 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0}\\ \end{array} \]

Alternative 5
Error	1.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4441594.753858191:\\ \;\;\;\;\frac{\beta + 1}{\left(4 + \beta \cdot \left(\beta + 4\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4441594.753858191:\\ \;\;\;\;\frac{\beta + 1}{\left(4 + \beta \cdot \left(\beta + 4\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

Alternative 7
Error	2.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.7314148262823104:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 8
Error	1.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.7314148262823104:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 9
Error	4.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.7314148262823104:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 6.096931076142862 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Error	3.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.7314148262823104:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 11
Error	5.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.7314148262823104:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12
Error	35.1
Cost	320

\[0.08333333333333333 + \alpha \cdot -0.027777777777777776 \]

Alternative 13
Error	35.2
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2022316 
(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)))