Octave 3.8, jcobi/3

Average Error: 3.6 → 0.2

Time: 20.1s

Precision: binary64

Cost: 1732

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)} \cdot \frac{-1 - \beta}{t_0 \cdot \left(-2 - \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{t_0}\\ \end{array} \]

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 3.0))))
   (if (<= beta 2.6e+15)
     (*
      (/ (+ alpha 1.0) (+ beta (+ alpha 2.0)))
      (/ (- -1.0 beta) (* t_0 (- -2.0 (+ alpha beta)))))
     (/
      (*
       (/ (+ alpha 1.0) (+ 2.0 (+ alpha beta)))
       (+ 1.0 (/ (- -1.0 alpha) beta)))
      t_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 + 3.0);
	double tmp;
	if (beta <= 2.6e+15) {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) * ((-1.0 - beta) / (t_0 * (-2.0 - (alpha + beta))));
	} else {
		tmp = (((alpha + 1.0) / (2.0 + (alpha + beta))) * (1.0 + ((-1.0 - alpha) / beta))) / t_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 + 3.0d0)
    if (beta <= 2.6d+15) then
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 2.0d0))) * (((-1.0d0) - beta) / (t_0 * ((-2.0d0) - (alpha + beta))))
    else
        tmp = (((alpha + 1.0d0) / (2.0d0 + (alpha + beta))) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / t_0
    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 + 3.0);
	double tmp;
	if (beta <= 2.6e+15) {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) * ((-1.0 - beta) / (t_0 * (-2.0 - (alpha + beta))));
	} else {
		tmp = (((alpha + 1.0) / (2.0 + (alpha + beta))) * (1.0 + ((-1.0 - alpha) / beta))) / t_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 + 3.0)
	tmp = 0
	if beta <= 2.6e+15:
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) * ((-1.0 - beta) / (t_0 * (-2.0 - (alpha + beta))))
	else:
		tmp = (((alpha + 1.0) / (2.0 + (alpha + beta))) * (1.0 + ((-1.0 - alpha) / beta))) / t_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(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 2.6e+15)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 2.0))) * Float64(Float64(-1.0 - beta) / Float64(t_0 * Float64(-2.0 - Float64(alpha + beta)))));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(alpha + beta))) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / t_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 + 3.0);
	tmp = 0.0;
	if (beta <= 2.6e+15)
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) * ((-1.0 - beta) / (t_0 * (-2.0 - (alpha + beta))));
	else
		tmp = (((alpha + 1.0) / (2.0 + (alpha + beta))) * (1.0 + ((-1.0 - alpha) / beta))) / t_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[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.6e+15], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 - beta), $MachinePrecision] / N[(t$95$0 * N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6e15

   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

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)}} \]
      Proof
      (/.f64 (/.f64 (+.f64 alpha 1) (*.f64 (+.f64 alpha (+.f64 beta 2)) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 alpha)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2)) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 13 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 13 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (/.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 1 alpha) (/.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 beta 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (+.f64 alpha (+.f64 beta 3))): 13 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 13 points decrease in error
      (/.f64 (/.f64 (/.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 alpha 1)) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 alpha (+.f64 beta 1)) (+.f64 beta 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 13 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 beta 1) alpha)) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 13 points decrease in error
      (/.f64 (/.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 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 beta alpha) (+.f64 alpha (+.f64 beta 1)))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 beta alpha) (+.f64 alpha beta)) 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha))) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.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 alpha (+.f64 beta (Rewrite<= metadata-eval (+.f64 2 1))))): 20 points increase in error, 0 points decrease in error
      (/.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))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (+.f64 2 1)))): 0 points increase in error, 20 points decrease in error
      (/.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))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) 2) 1))): 0 points increase in error, 0 points decrease in error
      (/.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) (Rewrite<= metadata-eval (*.f64 2 1))) 1)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

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

   5. Simplified 0.1

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{-1 - \beta}{\left(\left(\beta + 3\right) + \alpha\right) \cdot \left(-2 - \left(\beta + \alpha\right)\right)}} \]
      Proof
      (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 2 alpha) beta)) (/.f64 (-.f64 -1 beta) (*.f64 (+.f64 (+.f64 beta 3) alpha) (-.f64 -2 (+.f64 beta alpha))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha 1)) (+.f64 (+.f64 2 alpha) beta)) (/.f64 (-.f64 -1 beta) (*.f64 (+.f64 (+.f64 beta 3) alpha) (-.f64 -2 (+.f64 beta alpha))))): 0 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 alpha 2)) beta)) (/.f64 (-.f64 -1 beta) (*.f64 (+.f64 (+.f64 beta 3) alpha) (-.f64 -2 (+.f64 beta alpha))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 alpha 1) (Rewrite<= associate-+r+_binary64 (+.f64 alpha (+.f64 2 beta)))) (/.f64 (-.f64 -1 beta) (*.f64 (+.f64 (+.f64 beta 3) alpha) (-.f64 -2 (+.f64 beta alpha))))): 6 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 2 beta))) (/.f64 (-.f64 -1 beta) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta 3))) (-.f64 -2 (+.f64 beta alpha))))): 0 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 2 beta))) (/.f64 (-.f64 -1 beta) (*.f64 (+.f64 alpha (+.f64 beta 3)) (-.f64 -2 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)))))): 9 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 2 beta))) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (-.f64 -1 beta) (-.f64 -2 (+.f64 alpha beta))) (+.f64 alpha (+.f64 beta 3))))): 0 points increase in error, 9 points decrease in error
      (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 2 beta))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (-.f64 -1 beta) (-.f64 -2 (+.f64 alpha beta))) 1)) (+.f64 alpha (+.f64 beta 3)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 2 beta))) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (-.f64 -1 beta) (-.f64 -2 (+.f64 alpha beta))) (/.f64 1 (+.f64 alpha (+.f64 beta 3)))))): 0 points increase in error, 0 points decrease in error

   if 2.6e15 < beta

   1. Initial program 6.6

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)}} \]
      Proof
      (/.f64 (/.f64 (+.f64 alpha 1) (*.f64 (+.f64 alpha (+.f64 beta 2)) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 alpha)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2)) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (/.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 13 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2)) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 13 points decrease in error
      (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (/.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (+.f64 beta 1)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 1 alpha) (/.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 beta 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (+.f64 alpha (+.f64 beta 3))): 13 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 13 points decrease in error
      (/.f64 (/.f64 (/.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 alpha 1)) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 alpha (+.f64 beta 1)) (+.f64 beta 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 13 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 beta 1) alpha)) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 13 points decrease in error
      (/.f64 (/.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 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 beta alpha) (+.f64 alpha (+.f64 beta 1)))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 beta alpha) (+.f64 alpha beta)) 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha))) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.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 alpha (+.f64 beta (Rewrite<= metadata-eval (+.f64 2 1))))): 20 points increase in error, 0 points decrease in error
      (/.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))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (+.f64 2 1)))): 0 points increase in error, 20 points decrease in error
      (/.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))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) 2) 1))): 0 points increase in error, 0 points decrease in error
      (/.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) (Rewrite<= metadata-eval (*.f64 2 1))) 1)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.2

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

   5. Simplified 0.2

      \[\leadsto \frac{\color{blue}{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \left(\left(\beta + 1\right) \cdot \frac{1}{2 + \left(\alpha + \beta\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]
      Proof
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 2 (+.f64 alpha beta))) (*.f64 (+.f64 beta 1) (/.f64 1 (+.f64 2 (+.f64 alpha beta))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 2 alpha) beta))) (*.f64 (+.f64 beta 1) (/.f64 1 (+.f64 2 (+.f64 alpha beta))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (Rewrite<= +-commutative_binary64 (+.f64 beta (+.f64 2 alpha)))) (*.f64 (+.f64 beta 1) (/.f64 1 (+.f64 2 (+.f64 alpha beta))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 beta (+.f64 2 alpha))) (*.f64 (+.f64 beta 1) (/.f64 1 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 2 alpha) beta))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 beta (+.f64 2 alpha))) (*.f64 (+.f64 beta 1) (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 beta (+.f64 2 alpha)))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 beta (+.f64 2 alpha))) (+.f64 beta 1)) (/.f64 1 (+.f64 beta (+.f64 2 alpha))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 alpha)) (+.f64 beta (+.f64 2 alpha))) (+.f64 beta 1)) (/.f64 1 (+.f64 beta (+.f64 2 alpha)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 beta (+.f64 2 alpha))) (Rewrite=> +-commutative_binary64 (+.f64 1 beta))) (/.f64 1 (+.f64 beta (+.f64 2 alpha)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 beta (+.f64 2 alpha))) (+.f64 (Rewrite<= metadata-eval (-.f64 0 -1)) beta)) (/.f64 1 (+.f64 beta (+.f64 2 alpha)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 beta (+.f64 2 alpha))) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 -1 beta)))) (/.f64 1 (+.f64 beta (+.f64 2 alpha)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 beta (+.f64 2 alpha))) (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 -1 beta)))) (/.f64 1 (+.f64 beta (+.f64 2 alpha)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 1 alpha) (/.f64 (+.f64 beta (+.f64 2 alpha)) (neg.f64 (-.f64 -1 beta))))) (/.f64 1 (+.f64 beta (+.f64 2 alpha)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 alpha 1)) (/.f64 (+.f64 beta (+.f64 2 alpha)) (neg.f64 (-.f64 -1 beta)))) (/.f64 1 (+.f64 beta (+.f64 2 alpha)))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 (+.f64 beta (+.f64 2 alpha))) (/.f64 (+.f64 alpha 1) (/.f64 (+.f64 beta (+.f64 2 alpha)) (neg.f64 (-.f64 -1 beta)))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in beta around inf 0.3

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

   7. Simplified 0.3

      \[\leadsto \frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \color{blue}{\left(1 + \frac{-1 - \alpha}{\beta}\right)}}{\alpha + \left(\beta + 3\right)} \]
      Proof
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 2 (+.f64 alpha beta))) (+.f64 1 (/.f64 (-.f64 -1 alpha) beta))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 2 (+.f64 alpha beta))) (+.f64 1 (/.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 0 1)) alpha) beta))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 2 (+.f64 alpha beta))) (+.f64 1 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.f64 1 alpha))) beta))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 2 (+.f64 alpha beta))) (+.f64 1 (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 1 alpha))) beta))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 2 (+.f64 alpha beta))) (+.f64 1 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 1 alpha))) beta))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 2 (+.f64 alpha beta))) (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 1 alpha) beta))))) (+.f64 alpha (+.f64 beta 3))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)} \cdot \frac{-1 - \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(-2 - \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1600

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

Alternative 3
Error	0.1
Cost	1600

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

Alternative 4
Error	0.9
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 5
Error	1.4
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta + \left(\alpha + 2\right)}}{\left(-3 - \beta\right) - \alpha}\\ \end{array} \]

Alternative 6
Error	1.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;\frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	1092

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

Alternative 8
Error	1.8
Cost	836

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

Alternative 9
Error	1.8
Cost	836

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

Alternative 10
Error	24.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 11
Error	2.1
Cost	708

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

Alternative 12
Error	27.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 5\right)}\\ \end{array} \]

Alternative 13
Error	25.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14
Error	56.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \]

Alternative 15
Error	38.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Error	27.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 17
Error	56.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \]

Alternative 18
Error	57.2
Cost	64

\[0.16666666666666666 \]

Reproduce

herbie shell --seed 2022343 
(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)))