Octave 3.8, jcobi/3

Average Error: 3.8 → 0.3

Time: 27.0s

Precision: binary64

Cost: 4804

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \left(\alpha + \beta\right) + 3\\ t_2 := \frac{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{t_0}}{t_0}\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{t_2 \cdot \left(t_2 \cdot \frac{1}{t_2}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{t_1}\\ \end{array} \]

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)))
        (t_1 (+ (+ alpha beta) 3.0))
        (t_2 (/ (/ (+ alpha (+ beta (+ (* alpha beta) 1.0))) t_0) t_0)))
   (if (<= beta 2.5e+56)
     (/ (* t_2 (* t_2 (/ 1.0 t_2))) t_1)
     (/ (+ (/ 1.0 beta) (/ alpha beta)) t_1))))

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 t_1 = (alpha + beta) + 3.0;
	double t_2 = ((alpha + (beta + ((alpha * beta) + 1.0))) / t_0) / t_0;
	double tmp;
	if (beta <= 2.5e+56) {
		tmp = (t_2 * (t_2 * (1.0 / t_2))) / t_1;
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (alpha + beta) + 3.0d0
    t_2 = ((alpha + (beta + ((alpha * beta) + 1.0d0))) / t_0) / t_0
    if (beta <= 2.5d+56) then
        tmp = (t_2 * (t_2 * (1.0d0 / t_2))) / t_1
    else
        tmp = ((1.0d0 / beta) + (alpha / beta)) / t_1
    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 t_1 = (alpha + beta) + 3.0;
	double t_2 = ((alpha + (beta + ((alpha * beta) + 1.0))) / t_0) / t_0;
	double tmp;
	if (beta <= 2.5e+56) {
		tmp = (t_2 * (t_2 * (1.0 / t_2))) / t_1;
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / t_1;
	}
	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)
	t_1 = (alpha + beta) + 3.0
	t_2 = ((alpha + (beta + ((alpha * beta) + 1.0))) / t_0) / t_0
	tmp = 0
	if beta <= 2.5e+56:
		tmp = (t_2 * (t_2 * (1.0 / t_2))) / t_1
	else:
		tmp = ((1.0 / beta) + (alpha / beta)) / t_1
	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))
	t_1 = Float64(Float64(alpha + beta) + 3.0)
	t_2 = Float64(Float64(Float64(alpha + Float64(beta + Float64(Float64(alpha * beta) + 1.0))) / t_0) / t_0)
	tmp = 0.0
	if (beta <= 2.5e+56)
		tmp = Float64(Float64(t_2 * Float64(t_2 * Float64(1.0 / t_2))) / t_1);
	else
		tmp = Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) / t_1);
	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);
	t_1 = (alpha + beta) + 3.0;
	t_2 = ((alpha + (beta + ((alpha * beta) + 1.0))) / t_0) / t_0;
	tmp = 0.0;
	if (beta <= 2.5e+56)
		tmp = (t_2 * (t_2 * (1.0 / t_2))) / t_1;
	else
		tmp = ((1.0 / beta) + (alpha / beta)) / t_1;
	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]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(alpha + N[(beta + N[(N[(alpha * beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2.5e+56], N[(N[(t$95$2 * N[(t$95$2 * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.50000000000000012e56

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{\frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.1	\[ \frac{\frac{\frac{\color{blue}{\left(\alpha + \beta\right) + \left(1 + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \color{blue}{\left(\beta \cdot \alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\color{blue}{\alpha \cdot \beta} + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\color{blue}{\alpha + \left(2 + \beta\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \color{blue}{\left(\beta + 2\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\alpha + \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{1 + \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \beta\right) + \left(1 + 2\right)}} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

   3. Applied egg-rr 0.1

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

   if 2.50000000000000012e56 < beta

   1. Initial program 7.8

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

   2. Simplified 7.8

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

      [Start] 7.8	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.8	\[ \frac{\frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 7.8	\[ \frac{\frac{\frac{\color{blue}{\left(\alpha + \beta\right) + \left(1 + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \color{blue}{\left(\beta \cdot \alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\color{blue}{\alpha \cdot \beta} + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      metadata-eval [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\color{blue}{\alpha + \left(2 + \beta\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \color{blue}{\left(\beta + 2\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      metadata-eval [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\alpha + \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      metadata-eval [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{1 + \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \beta\right) + \left(1 + 2\right)}} \]
      metadata-eval [=>] 7.8	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

   3. Taylor expanded in beta around inf 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.6	\[ \frac{\frac{\color{blue}{\alpha + 1}}{\beta}}{\left(\alpha + \beta\right) + 3} \]
      rational_best_oopsla_all_46_json_45_simplify-1 [<=] 0.6	\[ \frac{\frac{\color{blue}{\alpha - -1}}{\beta}}{\left(\alpha + \beta\right) + 3} \]

   5. Taylor expanded in alpha around 0 0.6

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \left(\frac{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\frac{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}\right)}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	1860

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{2 + \alpha}}{t_1}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\beta} + \left(\frac{\alpha}{\beta} + \left(1 + \alpha\right)\right)\right) - \frac{2}{\beta}}{t_1}}{t_0}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1860

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{t_1}}{t_1}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 3
Error	1.1
Cost	1348

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 22000000000000:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 2}}{t_0}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_1}\\ \end{array} \]

Alternative 4
Error	1.5
Cost	1092

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{t_0}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	964

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 6
Error	1.6
Cost	836

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{t_0}\\ \end{array} \]

Alternative 7
Error	4.8
Cost	708

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{t_0}\\ \end{array} \]

Alternative 8
Error	31.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{0.25}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \alpha}\\ \end{array} \]

Alternative 9
Error	5.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.25}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Error	34.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;0.08333333333333333 + -0.027777777777777776 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 11
Error	32.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+139}:\\ \;\;\;\;\frac{0.25}{3 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \alpha}\\ \end{array} \]

Alternative 12
Error	35.2
Cost	320

\[\frac{0.25}{3 + \alpha} \]

Alternative 13
Error	35.6
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023090 
(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)))