Octave 3.8, jcobi/3

Average Accuracy: 94.4% → 99.2%

Time: 30.4s

Precision: binary64

Cost: 1608

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 3}}{t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

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 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.75e-106)
     (/ (/ (+ alpha 1.0) (+ alpha 3.0)) (* t_0 t_0))
     (if (<= beta 4.2e+16)
       (/
        1.0
        (*
         (+ (/ beta (+ 1.0 beta)) (/ 2.0 (+ 1.0 beta)))
         (* (+ beta 2.0) (+ beta 3.0))))
       (/
        (/ (+ alpha 1.0) (+ (+ beta 3.0) (* alpha 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 (beta <= 1.75e-106) {
		tmp = ((alpha + 1.0) / (alpha + 3.0)) / (t_0 * t_0);
	} else if (beta <= 4.2e+16) {
		tmp = 1.0 / (((beta / (1.0 + beta)) + (2.0 / (1.0 + beta))) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 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 (beta <= 1.75d-106) then
        tmp = ((alpha + 1.0d0) / (alpha + 3.0d0)) / (t_0 * t_0)
    else if (beta <= 4.2d+16) then
        tmp = 1.0d0 / (((beta / (1.0d0 + beta)) + (2.0d0 / (1.0d0 + beta))) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / ((beta + 3.0d0) + (alpha * 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 (beta <= 1.75e-106) {
		tmp = ((alpha + 1.0) / (alpha + 3.0)) / (t_0 * t_0);
	} else if (beta <= 4.2e+16) {
		tmp = 1.0 / (((beta / (1.0 + beta)) + (2.0 / (1.0 + beta))) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 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 beta <= 1.75e-106:
		tmp = ((alpha + 1.0) / (alpha + 3.0)) / (t_0 * t_0)
	elif beta <= 4.2e+16:
		tmp = 1.0 / (((beta / (1.0 + beta)) + (2.0 / (1.0 + beta))) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 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(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.75e-106)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + 3.0)) / Float64(t_0 * t_0));
	elseif (beta <= 4.2e+16)
		tmp = Float64(1.0 / Float64(Float64(Float64(beta / Float64(1.0 + beta)) + Float64(2.0 / Float64(1.0 + beta))) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + 3.0) + Float64(alpha * 2.0))) / 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 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.75e-106)
		tmp = ((alpha + 1.0) / (alpha + 3.0)) / (t_0 * t_0);
	elseif (beta <= 4.2e+16)
		tmp = 1.0 / (((beta / (1.0 + beta)) + (2.0 / (1.0 + beta))) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.75e-106], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.2e+16], N[(1.0 / N[(N[(N[(beta / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.75e-106

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 99.9	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/r* [<=] 99.9	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
      associate-/l/ [<=] 99.9	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Taylor expanded in beta around 0 99.9%

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

   4. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{\frac{1 + \alpha}{3 + \alpha}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      +-commutative [<=] 99.9	\[ \frac{\frac{1 + \alpha}{\color{blue}{\alpha + 3}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   if 1.75e-106 < beta < 4.2e16

   1. Initial program 99.7%

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

   2. Simplified 99.7%

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

      [Start] 99.7	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Applied egg-rr 99.7%

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

      [Start] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)} \]
      div-inv [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{1}{\beta + 1}\right)}}}{\alpha + \left(\beta + 3\right)} \]
      *-commutative [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\frac{1}{\beta + 1} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}}{\alpha + \left(\beta + 3\right)} \]
      frac-2neg [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\color{blue}{\frac{-1}{-\left(\beta + 1\right)}} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{\color{blue}{-1}}{-\left(\beta + 1\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      neg-sub0 [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{\color{blue}{0 - \left(\beta + 1\right)}} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [<=] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{\color{blue}{\log 1} - \left(\beta + 1\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      +-commutative [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{\log 1 - \color{blue}{\left(1 + \beta\right)}} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      associate--r+ [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{\color{blue}{\left(\log 1 - 1\right) - \beta}} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{\left(\color{blue}{0} - 1\right) - \beta} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{\color{blue}{-1} - \beta} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      associate-+r+ [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{-1 - \beta} \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]
      +-commutative [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{-1 - \beta} \cdot \color{blue}{\left(2 + \left(\alpha + \beta\right)\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]

   4. Applied egg-rr 99.7%

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

      [Start] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-1}{-1 - \beta} \cdot \left(2 + \left(\alpha + \beta\right)\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      distribute-rgt-in [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(2 \cdot \frac{-1}{-1 - \beta} + \left(\alpha + \beta\right) \cdot \frac{-1}{-1 - \beta}\right)}}}{\alpha + \left(\beta + 3\right)} \]
      associate-*r/ [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\color{blue}{\frac{2 \cdot -1}{-1 - \beta}} + \left(\alpha + \beta\right) \cdot \frac{-1}{-1 - \beta}\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{\color{blue}{-2}}{-1 - \beta} + \left(\alpha + \beta\right) \cdot \frac{-1}{-1 - \beta}\right)}}{\alpha + \left(\beta + 3\right)} \]
      frac-2neg [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \left(\alpha + \beta\right) \cdot \color{blue}{\frac{--1}{-\left(-1 - \beta\right)}}\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \left(\alpha + \beta\right) \cdot \frac{\color{blue}{1}}{-\left(-1 - \beta\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]
      un-div-inv [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \color{blue}{\frac{\alpha + \beta}{-\left(-1 - \beta\right)}}\right)}}{\alpha + \left(\beta + 3\right)} \]
      +-commutative [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \frac{\color{blue}{\beta + \alpha}}{-\left(-1 - \beta\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]

   5. Simplified 99.7%

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

      [Start] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \frac{\beta + \alpha}{-\left(-1 - \beta\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]
      +-commutative [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \frac{\color{blue}{\alpha + \beta}}{-\left(-1 - \beta\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]
      neg-sub0 [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \frac{\alpha + \beta}{\color{blue}{0 - \left(-1 - \beta\right)}}\right)}}{\alpha + \left(\beta + 3\right)} \]
      associate--r- [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \frac{\alpha + \beta}{\color{blue}{\left(0 - -1\right) + \beta}}\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \frac{\alpha + \beta}{\color{blue}{1} + \beta}\right)}}{\alpha + \left(\beta + 3\right)} \]
      +-commutative [<=] 99.7	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\frac{-2}{-1 - \beta} + \frac{\alpha + \beta}{\color{blue}{\beta + 1}}\right)}}{\alpha + \left(\beta + 3\right)} \]

   6. Taylor expanded in alpha around 0 97.9%

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

   7. Simplified 97.9%

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

      [Start] 97.9	\[ \frac{1}{\left(\frac{\beta}{\beta + 1} + 2 \cdot \frac{1}{\beta + 1}\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)} \]
      +-commutative [=>] 97.9	\[ \frac{1}{\left(\frac{\beta}{\color{blue}{1 + \beta}} + 2 \cdot \frac{1}{\beta + 1}\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)} \]
      associate-*r/ [=>] 97.9	\[ \frac{1}{\left(\frac{\beta}{1 + \beta} + \color{blue}{\frac{2 \cdot 1}{\beta + 1}}\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)} \]
      metadata-eval [=>] 97.9	\[ \frac{1}{\left(\frac{\beta}{1 + \beta} + \frac{\color{blue}{2}}{\beta + 1}\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)} \]
      +-commutative [=>] 97.9	\[ \frac{1}{\left(\frac{\beta}{1 + \beta} + \frac{2}{\color{blue}{1 + \beta}}\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)} \]

   if 4.2e16 < beta

   1. Initial program 89.6%

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

   2. Simplified 99.8%

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

      [Start] 89.6	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Taylor expanded in beta around inf 99.3%

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

   4. Simplified 99.3%

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

      [Start] 99.3	\[ \frac{\frac{\alpha + 1}{\beta + \left(3 + 2 \cdot \alpha\right)}}{\alpha + \left(\beta + 3\right)} \]
      associate-+r+ [=>] 99.3	\[ \frac{\frac{\alpha + 1}{\color{blue}{\left(\beta + 3\right) + 2 \cdot \alpha}}}{\alpha + \left(\beta + 3\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 3}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1856

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

Alternative 2
Accuracy	99.8%
Cost	1728

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

Alternative 3
Accuracy	99.5%
Cost	1604

\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 62000000:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{t_0 \cdot \left(t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	1600

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

Alternative 5
Accuracy	99.8%
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 6
Accuracy	99.3%
Cost	1480

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 3}}{t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 13500000:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Accuracy	98.6%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 56000000:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8
Accuracy	97.6%
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 1:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha \cdot 2 + 4}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 9
Accuracy	96.8%
Cost	1092

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha \cdot 2 + 4}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0}\\ \end{array} \]

Alternative 10
Accuracy	97.4%
Cost	1092

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 1:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha \cdot 2 + 4}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{t_0}\\ \end{array} \]

Alternative 11
Accuracy	96.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 12
Accuracy	91.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 13
Accuracy	93.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14
Accuracy	96.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 15
Accuracy	90.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Accuracy	46.4%
Cost	320

\[\frac{0.16666666666666666}{\beta + 2} \]

Alternative 17
Accuracy	2.5%
Cost	192

\[\frac{0.5}{\alpha} \]

Reproduce

herbie shell --seed 2023126 
(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)))