Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.6%

Time: 22.5s

Alternatives: 21

Speedup: 2.9×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.15e+51)
     (* (+ 1.0 beta) (/ (+ 1.0 alpha) (* (+ alpha (+ beta 3.0)) (* t_0 t_0))))
     (/
      (* (+ 1.0 alpha) (/ 1.0 (+ 4.0 (+ beta (* alpha 2.0)))))
      (+ beta (+ alpha 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.15e+51) {
		tmp = (1.0 + beta) * ((1.0 + alpha) / ((alpha + (beta + 3.0)) * (t_0 * t_0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this 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 <= 2.15d+51) then
        tmp = (1.0d0 + beta) * ((1.0d0 + alpha) / ((alpha + (beta + 3.0d0)) * (t_0 * t_0)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / (4.0d0 + (beta + (alpha * 2.0d0))))) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.15e+51) {
		tmp = (1.0 + beta) * ((1.0 + alpha) / ((alpha + (beta + 3.0)) * (t_0 * t_0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.15e+51:
		tmp = (1.0 + beta) * ((1.0 + alpha) / ((alpha + (beta + 3.0)) * (t_0 * t_0)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.15e+51)
		tmp = Float64(Float64(1.0 + beta) * Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(4.0 + Float64(beta + Float64(alpha * 2.0))))) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.15e+51)
		tmp = (1.0 + beta) * ((1.0 + alpha) / ((alpha + (beta + 3.0)) * (t_0 * t_0)));
	else
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.15e+51], N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(4.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.1499999999999999e51

   1. Initial program 99.8%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 99.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. expm1-def 94.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\frac{\left(\alpha + \left(\beta + 3\right)\right) \cdot {\left(\beta + \left(\alpha + 2\right)\right)}^{2}}{\beta + 1}}\right)\right)} \]

      2. expm1-log1p 94.6%

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

      3. associate-/r/ 94.6%

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

      4. +-commutative 94.6%

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

      5. +-commutative 94.6%

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

      6. +-commutative 94.6%

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

      7. +-commutative 94.6%

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

   8. Simplified 94.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \left(3 + \beta\right)\right) \cdot {\left(\left(2 + \alpha\right) + \beta\right)}^{2}} \cdot \left(1 + \beta\right)} \]
   9. Step-by-step derivation

      1. +-commutative 94.6%

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

      2. +-commutative 94.6%

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

      3. pow2 94.6%

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

      4. associate-+r+ 94.6%

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

      5. +-commutative 94.6%

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

      6. associate-+r+ 94.6%

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

      7. +-commutative 94.6%

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

      8. associate-+r+ 94.6%

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

      9. associate-+r+ 94.6%

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

   10. Applied egg-rr 94.6%

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

   if 2.1499999999999999e51 < beta

   1. Initial program 87.0%

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

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 92.4%

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

      2. +-commutative 92.4%

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

      3. associate-+r+ 92.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}} \]
   7. Step-by-step derivation

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\left(1 + \alpha\right) \cdot \color{blue}{{\left(\frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}}}{\beta + \left(\alpha + 2\right)} \]
   9. Step-by-step derivation

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in beta around inf 87.8%

       \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\color{blue}{4 + \left(\beta + 2 \cdot \alpha\right)}}}{\beta + \left(\alpha + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 49000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{t\_0}}{t\_0 \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 49000000.0)
     (/ (/ (+ 1.0 (+ alpha beta)) t_0) (* t_0 (+ 3.0 (+ alpha beta))))
     (/
      (* (+ 1.0 alpha) (/ 1.0 (+ 4.0 (+ beta (* alpha 2.0)))))
      (+ beta (+ alpha 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 49000000.0) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this 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 <= 49000000.0d0) then
        tmp = ((1.0d0 + (alpha + beta)) / t_0) / (t_0 * (3.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / (4.0d0 + (beta + (alpha * 2.0d0))))) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 49000000.0) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 49000000.0:
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 49000000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + beta)) / t_0) / Float64(t_0 * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(4.0 + Float64(beta + Float64(alpha * 2.0))))) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 49000000.0)
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 49000000.0], N[(N[(N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(4.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9e7

   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. Step-by-step derivation

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.4%

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

      4. *-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. +-commutative 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 98.6%

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

   if 4.9e7 < beta

   1. Initial program 88.7%

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

   3. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 93.4%

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

      2. +-commutative 93.4%

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

      3. associate-+r+ 93.4%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}} \]
   7. Step-by-step derivation

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{\left(1 + \alpha\right) \cdot \color{blue}{{\left(\frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}}}{\beta + \left(\alpha + 2\right)} \]
   9. Step-by-step derivation

      1. unpow-1 99.8%

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

      2. associate-/r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in beta around inf 83.8%

       \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\color{blue}{4 + \left(\beta + 2 \cdot \alpha\right)}}}{\beta + \left(\alpha + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 49000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (/
  (*
   (+ 1.0 alpha)
   (/ 1.0 (* (/ (+ alpha (+ beta 3.0)) (+ 1.0 beta)) (+ alpha (+ beta 2.0)))))
  (+ beta (+ alpha 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((1.0 + alpha) * (1.0 / (((alpha + (beta + 3.0)) / (1.0 + beta)) * (alpha + (beta + 2.0))))) / (beta + (alpha + 2.0));
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((1.0d0 + alpha) * (1.0d0 / (((alpha + (beta + 3.0d0)) / (1.0d0 + beta)) * (alpha + (beta + 2.0d0))))) / (beta + (alpha + 2.0d0))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return ((1.0 + alpha) * (1.0 / (((alpha + (beta + 3.0)) / (1.0 + beta)) * (alpha + (beta + 2.0))))) / (beta + (alpha + 2.0));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return ((1.0 + alpha) * (1.0 / (((alpha + (beta + 3.0)) / (1.0 + beta)) * (alpha + (beta + 2.0))))) / (beta + (alpha + 2.0))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(1.0 + beta)) * Float64(alpha + Float64(beta + 2.0))))) / Float64(beta + Float64(alpha + 2.0)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((1.0 + alpha) * (1.0 / (((alpha + (beta + 3.0)) / (1.0 + beta)) * (alpha + (beta + 2.0))))) / (beta + (alpha + 2.0));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. +-commutative 97.5%

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

   3. associate-+r+ 97.5%

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

   4. associate-/r* 99.9%

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

   5. associate-+r+ 99.9%

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

   6. +-commutative 99.9%

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

   7. associate-+r+ 99.9%

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

   8. associate-+r+ 99.9%

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

   9. associate-+r+ 99.9%

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

   10. +-commutative 99.9%

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

   11. associate-+r+ 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}} \]
7. Step-by-step derivation

   1. clear-num 99.5%

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

   2. inv-pow 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-+r+ 99.5%

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

   5. +-commutative 99.5%

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

   6. associate-+r+ 99.5%

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

8. Applied egg-rr 99.5%

   \[\leadsto \frac{\left(1 + \alpha\right) \cdot \color{blue}{{\left(\frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}}}{\beta + \left(\alpha + 2\right)} \]
9. Step-by-step derivation

   1. unpow-1 99.5%

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

   2. associate-/r/ 99.5%

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

   3. +-commutative 99.5%

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

   4. +-commutative 99.5%

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

   5. +-commutative 99.5%

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

   6. +-commutative 99.5%

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

   7. +-commutative 99.5%

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

   8. +-commutative 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

    \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\frac{\alpha + \left(\beta + 3\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 2\right)\right)}}{\beta + \left(\alpha + 2\right)} \]
12. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (* (/ (/ (+ 1.0 alpha) t_0) t_0) (/ (+ 1.0 beta) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / (alpha + (beta + 3.0)));
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = beta + (alpha + 2.0d0)
    code = (((1.0d0 + alpha) / t_0) / t_0) * ((1.0d0 + beta) / (alpha + (beta + 3.0d0)))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / (alpha + (beta + 3.0)));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	return (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / (alpha + (beta + 3.0)))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	return Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0) * Float64(Float64(1.0 + beta) / Float64(alpha + Float64(beta + 3.0))))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / (alpha + (beta + 3.0)));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.4%

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

3. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 97.5%

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

   2. +-commutative 97.5%

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

   3. associate-+r+ 97.5%

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

   4. +-commutative 97.5%

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

   5. associate-+r+ 97.5%

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

   6. associate-+r+ 97.5%

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

   7. +-commutative 97.5%

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

   8. associate-+r+ 97.5%

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

6. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
7. Step-by-step derivation

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \]
10. Add Preprocessing

Alternative 5: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t\_0}}{\alpha + \left(\beta + 3\right)}}{t\_0} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (/ (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_0) (+ alpha (+ beta 3.0)))) t_0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return ((1.0 + alpha) * (((1.0 + beta) / t_0) / (alpha + (beta + 3.0)))) / t_0;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = beta + (alpha + 2.0d0)
    code = ((1.0d0 + alpha) * (((1.0d0 + beta) / t_0) / (alpha + (beta + 3.0d0)))) / t_0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return ((1.0 + alpha) * (((1.0 + beta) / t_0) / (alpha + (beta + 3.0)))) / t_0;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	return ((1.0 + alpha) * (((1.0 + beta) / t_0) / (alpha + (beta + 3.0)))) / t_0

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	return Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / t_0) / Float64(alpha + Float64(beta + 3.0)))) / t_0)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = ((1.0 + alpha) * (((1.0 + beta) / t_0) / (alpha + (beta + 3.0)))) / t_0;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 96.4%

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

3. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. +-commutative 97.5%

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

   3. associate-+r+ 97.5%

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

   4. associate-/r* 99.9%

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

   5. associate-+r+ 99.9%

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

   6. +-commutative 99.9%

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

   7. associate-+r+ 99.9%

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

   8. associate-+r+ 99.9%

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

   9. associate-+r+ 99.9%

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

   10. +-commutative 99.9%

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

   11. associate-+r+ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)} \]
8. Add Preprocessing

Alternative 6: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta} \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5e+29)
   (/
    1.0
    (* (/ (+ alpha (+ beta 2.0)) (+ 1.0 beta)) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (* (+ 1.0 alpha) (/ 1.0 beta)) (+ beta (+ alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5e+29) {
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d+29) then
        tmp = 1.0d0 / (((alpha + (beta + 2.0d0)) / (1.0d0 + beta)) * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5e+29) {
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5e+29:
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5e+29)
		tmp = Float64(1.0 / Float64(Float64(Float64(alpha + Float64(beta + 2.0)) / Float64(1.0 + beta)) * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5e+29)
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5e+29], N[(1.0 / N[(N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5000000000000002e29

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.6%

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

      2. +-commutative 72.6%

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

   9. Simplified 72.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}}{\beta + \left(\alpha + 2\right)} \]
   10. Step-by-step derivation

       1. clear-num 72.6%

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

       2. inv-pow 72.6%

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

       3. +-commutative 72.6%

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

       4. associate-+l+ 72.6%

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

       5. +-commutative 72.6%

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

       6. *-commutative 72.6%

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

   11. Applied egg-rr 72.6%

       \[\leadsto \color{blue}{{\left(\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 72.6%

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

       2. associate-/r/ 72.6%

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

       3. +-commutative 72.6%

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

       4. +-commutative 72.6%

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

       5. +-commutative 72.6%

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

       6. +-commutative 72.6%

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

       7. +-commutative 72.6%

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

       8. *-commutative 72.6%

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

       9. +-commutative 72.6%

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

       10. +-commutative 72.6%

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

   13. Simplified 72.6%

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

   if 4.5000000000000002e29 < beta

   1. Initial program 88.5%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 93.2%

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

      2. +-commutative 93.2%

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

      3. associate-+r+ 93.2%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf 84.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta} \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 75000000:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta} \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 75000000.0)
   (/
    1.0
    (* (/ (+ alpha (+ beta 2.0)) (+ 1.0 beta)) (* (+ beta 3.0) (+ beta 2.0))))
   (/
    (* (+ 1.0 alpha) (/ 1.0 (+ 4.0 (+ beta (* alpha 2.0)))))
    (+ beta (+ alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 75000000.0) {
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 75000000.0d0) then
        tmp = 1.0d0 / (((alpha + (beta + 2.0d0)) / (1.0d0 + beta)) * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / (4.0d0 + (beta + (alpha * 2.0d0))))) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 75000000.0) {
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 75000000.0:
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 75000000.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(alpha + Float64(beta + 2.0)) / Float64(1.0 + beta)) * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(4.0 + Float64(beta + Float64(alpha * 2.0))))) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 75000000.0)
		tmp = 1.0 / (((alpha + (beta + 2.0)) / (1.0 + beta)) * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) * (1.0 / (4.0 + (beta + (alpha * 2.0))))) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 75000000.0], N[(1.0 / N[(N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(4.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5e7

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.8%

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

      2. +-commutative 72.8%

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

   9. Simplified 72.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}}{\beta + \left(\alpha + 2\right)} \]
   10. Step-by-step derivation

       1. clear-num 72.8%

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

       2. inv-pow 72.8%

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

       3. +-commutative 72.8%

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

       4. associate-+l+ 72.8%

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

       5. +-commutative 72.8%

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

       6. *-commutative 72.8%

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

   11. Applied egg-rr 72.8%

       \[\leadsto \color{blue}{{\left(\frac{\alpha + \left(\beta + 2\right)}{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 72.8%

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

       2. associate-/r/ 72.8%

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

       3. +-commutative 72.8%

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

       4. +-commutative 72.8%

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

       5. +-commutative 72.8%

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

       6. +-commutative 72.8%

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

       7. +-commutative 72.8%

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

       8. *-commutative 72.8%

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

       9. +-commutative 72.8%

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

       10. +-commutative 72.8%

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

   13. Simplified 72.8%

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

   if 7.5e7 < beta

   1. Initial program 88.7%

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

   3. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 93.4%

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

      2. +-commutative 93.4%

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

      3. associate-+r+ 93.4%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}} \]
   7. Step-by-step derivation

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{\left(1 + \alpha\right) \cdot \color{blue}{{\left(\frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}}}{\beta + \left(\alpha + 2\right)} \]
   9. Step-by-step derivation

      1. unpow-1 99.8%

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

      2. associate-/r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in beta around inf 83.8%

       \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\color{blue}{4 + \left(\beta + 2 \cdot \alpha\right)}}}{\beta + \left(\alpha + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 75000000:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta} \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{4 + \left(\beta + \alpha \cdot 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2e+29)
   (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (* (+ 1.0 alpha) (/ 1.0 beta)) (+ beta (+ alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2e+29) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d+29) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2e+29) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.2e+29:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2e+29)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2e+29)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.2e+29], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.2000000000000003e29

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.6%

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

      2. +-commutative 72.6%

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

   9. Simplified 72.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}}{\beta + \left(\alpha + 2\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 72.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}\right)\right)} \]

       2. expm1-udef 78.3%

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

       3. *-commutative 78.3%

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

       4. +-commutative 78.3%

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

       5. associate-+l+ 78.3%

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

       6. +-commutative 78.3%

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

   11. Applied egg-rr 78.3%

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

       1. expm1-def 72.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

       2. expm1-log1p 72.6%

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

       3. associate-/l/ 72.6%

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

       4. *-commutative 72.6%

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

       5. +-commutative 72.6%

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

       6. +-commutative 72.6%

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

       7. *-commutative 72.6%

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

       8. +-commutative 72.6%

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

       9. +-commutative 72.6%

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

       10. +-commutative 72.6%

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

       11. +-commutative 72.6%

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

       12. +-commutative 72.6%

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

   13. Simplified 72.6%

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

   if 4.2000000000000003e29 < beta

   1. Initial program 88.5%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 93.2%

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

      2. +-commutative 93.2%

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

      3. associate-+r+ 93.2%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf 84.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= beta 5e+28)
     (/ (/ (+ 1.0 beta) (* (+ beta 3.0) (+ beta 2.0))) t_0)
     (/ (* (+ 1.0 alpha) (/ 1.0 beta)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 5e+28) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (beta + 2.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (beta <= 5d+28) then
        tmp = ((1.0d0 + beta) / ((beta + 3.0d0) * (beta + 2.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 5e+28) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (beta + 2.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if beta <= 5e+28:
		tmp = ((1.0 + beta) / ((beta + 3.0) * (beta + 2.0))) / t_0
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (beta <= 5e+28)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (beta <= 5e+28)
		tmp = ((1.0 + beta) / ((beta + 3.0) * (beta + 2.0))) / t_0;
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+28], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999957e28

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.6%

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

      2. +-commutative 72.6%

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

   9. Simplified 72.6%

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

   if 4.99999999999999957e28 < beta

   1. Initial program 88.5%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 93.2%

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

      2. +-commutative 93.2%

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

      3. associate-+r+ 93.2%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf 84.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2e+29)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (* (+ 1.0 alpha) (/ 1.0 beta)) (+ beta (+ alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2e+29) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d+29) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2e+29) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.2e+29:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2e+29)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2e+29)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.2e+29], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.2000000000000003e29

   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. Step-by-step derivation

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.4%

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

      4. *-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. +-commutative 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 84.0%

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

   6. Taylor expanded in alpha around 0 71.9%

      \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 71.9%

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

      2. +-commutative 71.9%

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

   8. Simplified 71.9%

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

   if 4.2000000000000003e29 < beta

   1. Initial program 88.5%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 93.2%

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

      2. +-commutative 93.2%

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

      3. associate-+r+ 93.2%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf 84.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= beta 6.0)
     (/ (+ 0.16666666666666666 (* alpha 0.027777777777777776)) t_0)
     (/ (* (+ 1.0 alpha) (/ 1.0 beta)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 6.0) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0;
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (beta <= 6.0d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.027777777777777776d0)) / t_0
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 6.0) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0;
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if beta <= 6.0:
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.027777777777777776)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0;
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.0], N[(N[(0.16666666666666666 + N[(alpha * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in beta around 0 97.4%

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

   8. Taylor expanded in alpha around 0 71.3%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \alpha}}{\beta + \left(\alpha + 2\right)} \]
   9. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto \frac{0.16666666666666666 + \color{blue}{\alpha \cdot 0.027777777777777776}}{\beta + \left(\alpha + 2\right)} \]

   10. Simplified 71.3%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot 0.027777777777777776}}{\beta + \left(\alpha + 2\right)} \]

   if 6 < beta

   1. Initial program 89.1%

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

   3. Simplified 93.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 93.6%

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

      2. +-commutative 93.6%

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

      3. associate-+r+ 93.6%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf 81.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.16666666666666666 (+ beta (+ alpha 2.0)))
   (if (<= beta 1.35e+154)
     (/ 1.0 (* beta (+ beta 2.0)))
     (* (/ 1.0 beta) (/ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else if (beta <= 1.35e+154) {
		tmp = 1.0 / (beta * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.16666666666666666d0 / (beta + (alpha + 2.0d0))
    else if (beta <= 1.35d+154) then
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else if (beta <= 1.35e+154) {
		tmp = 1.0 / (beta * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0))
	elif beta <= 1.35e+154:
		tmp = 1.0 / (beta * (beta + 2.0))
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.16666666666666666 / Float64(beta + Float64(alpha + 2.0)));
	elseif (beta <= 1.35e+154)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	elseif (beta <= 1.35e+154)
		tmp = 1.0 / (beta * (beta + 2.0));
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.16666666666666666 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.3%

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

      2. +-commutative 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in beta around 0 71.1%

       \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\beta + \left(\alpha + 2\right)} \]

   if 6.20000000000000018 < beta < 1.35000000000000003e154

   1. Initial program 92.6%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in beta around inf 65.5%

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

   6. Taylor expanded in alpha around 0 63.4%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(2 + \beta\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 63.4%

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

   8. Simplified 63.4%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in beta around inf 97.6%

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

   6. Taylor expanded in beta around inf 97.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{1}{\beta} \]

   7. Taylor expanded in alpha around inf 96.4%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta}} \cdot \frac{1}{\beta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6)
   (/ 0.16666666666666666 (+ beta (+ alpha 2.0)))
   (if (<= beta 4.5e+159)
     (/ (/ 1.0 (+ beta 2.0)) beta)
     (* (/ 1.0 beta) (/ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else if (beta <= 4.5e+159) {
		tmp = (1.0 / (beta + 2.0)) / beta;
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.6d0) then
        tmp = 0.16666666666666666d0 / (beta + (alpha + 2.0d0))
    else if (beta <= 4.5d+159) then
        tmp = (1.0d0 / (beta + 2.0d0)) / beta
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else if (beta <= 4.5e+159) {
		tmp = (1.0 / (beta + 2.0)) / beta;
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0))
	elif beta <= 4.5e+159:
		tmp = (1.0 / (beta + 2.0)) / beta
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(0.16666666666666666 / Float64(beta + Float64(alpha + 2.0)));
	elseif (beta <= 4.5e+159)
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) / beta);
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6)
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	elseif (beta <= 4.5e+159)
		tmp = (1.0 / (beta + 2.0)) / beta;
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.6], N[(0.16666666666666666 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.5e+159], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.5999999999999996

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.3%

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

      2. +-commutative 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in beta around 0 71.1%

       \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\beta + \left(\alpha + 2\right)} \]

   if 6.5999999999999996 < beta < 4.50000000000000026e159

   1. Initial program 90.7%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in beta around inf 64.9%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]
   6. Step-by-step derivation

      1. un-div-inv 65.2%

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

      2. +-commutative 65.2%

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

   7. Applied egg-rr 65.2%

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

   8. Taylor expanded in alpha around 0 64.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{2 + \beta}}}{\beta} \]
   9. Step-by-step derivation

      1. +-commutative 64.6%

         \[\leadsto \frac{\frac{1}{\color{blue}{\beta + 2}}}{\beta} \]

   10. Simplified 64.6%

       \[\leadsto \frac{\color{blue}{\frac{1}{\beta + 2}}}{\beta} \]

   if 4.50000000000000026e159 < beta

   1. Initial program 87.3%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

   6. Taylor expanded in beta around inf 100.0%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{1}{\beta} \]

   7. Taylor expanded in alpha around inf 100.0%

      \[\leadsto \color{blue}{\frac{\alpha}{\beta}} \cdot \frac{1}{\beta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8)
   (/
    (+ 0.16666666666666666 (* alpha 0.027777777777777776))
    (+ beta (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.8d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.027777777777777776d0)) / (beta + (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.8:
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.027777777777777776)) / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.8)
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.8], N[(N[(0.16666666666666666 + N[(alpha * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.79999999999999982

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in beta around 0 97.4%

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

   8. Taylor expanded in alpha around 0 71.3%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \alpha}}{\beta + \left(\alpha + 2\right)} \]
   9. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto \frac{0.16666666666666666 + \color{blue}{\alpha \cdot 0.027777777777777776}}{\beta + \left(\alpha + 2\right)} \]

   10. Simplified 71.3%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot 0.027777777777777776}}{\beta + \left(\alpha + 2\right)} \]

   if 7.79999999999999982 < beta

   1. Initial program 89.1%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in beta around inf 81.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]
   6. Step-by-step derivation

      1. un-div-inv 81.5%

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

      2. +-commutative 81.5%

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

   7. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}} \]
   8. Step-by-step derivation

      1. clear-num 81.5%

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

      2. inv-pow 81.5%

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

   9. Applied egg-rr 81.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}\right)}^{-1}}}{\beta} \]
   10. Step-by-step derivation

       1. unpow-1 81.5%

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

       2. +-commutative 81.5%

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

       3. +-commutative 81.5%

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

       4. +-commutative 81.5%

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

   11. Simplified 81.5%

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

   12. Taylor expanded in beta around inf 81.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/
    (+ 0.16666666666666666 (* alpha 0.027777777777777776))
    (+ beta (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.027777777777777776d0)) / (beta + (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.027777777777777776)) / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (beta + (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(N[(0.16666666666666666 + N[(alpha * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in beta around 0 97.4%

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

   8. Taylor expanded in alpha around 0 71.3%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \alpha}}{\beta + \left(\alpha + 2\right)} \]
   9. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto \frac{0.16666666666666666 + \color{blue}{\alpha \cdot 0.027777777777777776}}{\beta + \left(\alpha + 2\right)} \]

   10. Simplified 71.3%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot 0.027777777777777776}}{\beta + \left(\alpha + 2\right)} \]

   if 6 < beta

   1. Initial program 89.1%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in beta around inf 81.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]
   6. Step-by-step derivation

      1. associate-*l/ 81.5%

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

      2. +-commutative 81.5%

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

   7. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ 81.5%

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

      2. *-rgt-identity 81.5%

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

      3. +-commutative 81.5%

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

      4. +-commutative 81.5%

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

      5. +-commutative 81.5%

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

   9. Simplified 81.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5e+41)
   (/ 0.16666666666666666 (+ beta 2.0))
   (* (/ 1.0 beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5e+41) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d+41) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5e+41) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5e+41:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5e+41)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5e+41)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5e+41], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.49999999999999975e41

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in beta around 0 94.1%

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

   8. Taylor expanded in alpha around 0 67.4%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 67.4%

         \[\leadsto \frac{0.16666666666666666}{\color{blue}{\beta + 2}} \]

   10. Simplified 67.4%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

   if 6.49999999999999975e41 < beta

   1. Initial program 87.8%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in beta around inf 85.8%

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

   6. Taylor expanded in beta around inf 85.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{1}{\beta} \]

   7. Taylor expanded in alpha around inf 55.9%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4e+41)
   (/ 0.16666666666666666 (+ beta (+ alpha 2.0)))
   (* (/ 1.0 beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4e+41) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.4d+41) then
        tmp = 0.16666666666666666d0 / (beta + (alpha + 2.0d0))
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4e+41) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4e+41:
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0))
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4e+41)
		tmp = Float64(0.16666666666666666 / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4e+41)
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.4e+41], N[(0.16666666666666666 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.40000000000000019e41

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.0%

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

      2. +-commutative 72.0%

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

   9. Simplified 72.0%

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

   10. Taylor expanded in beta around 0 68.1%

       \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\beta + \left(\alpha + 2\right)} \]

   if 6.40000000000000019e41 < beta

   1. Initial program 87.8%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in beta around inf 85.8%

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

   6. Taylor expanded in beta around inf 85.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{1}{\beta} \]

   7. Taylor expanded in alpha around inf 55.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 96.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 0.16666666666666666 (+ beta (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.0d0) then
        tmp = 0.16666666666666666d0 / (beta + (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(0.16666666666666666 / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.0)
		tmp = 0.16666666666666666 / (beta + (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.0], N[(0.16666666666666666 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r+ 99.4%

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

      4. associate-/r* 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. associate-+r+ 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in alpha around 0 72.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative 72.3%

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

      2. +-commutative 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in beta around 0 71.1%

       \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\beta + \left(\alpha + 2\right)} \]

   if 8 < beta

   1. Initial program 89.1%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in beta around inf 81.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]
   6. Step-by-step derivation

      1. un-div-inv 81.5%

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

      2. +-commutative 81.5%

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

   7. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}} \]
   8. Step-by-step derivation

      1. clear-num 81.5%

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

      2. inv-pow 81.5%

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

   9. Applied egg-rr 81.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}\right)}^{-1}}}{\beta} \]
   10. Step-by-step derivation

       1. unpow-1 81.5%

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

       2. +-commutative 81.5%

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

       3. +-commutative 81.5%

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

       4. +-commutative 81.5%

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

   11. Simplified 81.5%

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

   12. Taylor expanded in beta around inf 81.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\alpha + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ alpha 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (alpha + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (alpha + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(alpha + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (alpha + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. +-commutative 97.5%

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

   3. associate-+r+ 97.5%

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

   4. associate-/r* 99.9%

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

   5. associate-+r+ 99.9%

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

   6. +-commutative 99.9%

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

   7. associate-+r+ 99.9%

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

   8. associate-+r+ 99.9%

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

   9. associate-+r+ 99.9%

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

   10. +-commutative 99.9%

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

   11. associate-+r+ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in alpha around 0 75.6%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\beta + \left(\alpha + 2\right)} \]
8. Step-by-step derivation

   1. +-commutative 75.6%

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

   2. +-commutative 75.6%

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

9. Simplified 75.6%

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

10. Taylor expanded in beta around 0 49.7%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

11. Final simplification 49.7%

    \[\leadsto \frac{0.16666666666666666}{\alpha + 2} \]
12. Add Preprocessing

Alternative 20: 46.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\beta + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ beta 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (beta + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (beta + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(beta + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (beta + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. +-commutative 97.5%

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

   3. associate-+r+ 97.5%

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

   4. associate-/r* 99.9%

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

   5. associate-+r+ 99.9%

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

   6. +-commutative 99.9%

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

   7. associate-+r+ 99.9%

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

   8. associate-+r+ 99.9%

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

   9. associate-+r+ 99.9%

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

   10. +-commutative 99.9%

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

   11. associate-+r+ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in beta around 0 71.0%

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

8. Taylor expanded in alpha around 0 49.7%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
9. Step-by-step derivation

   1. +-commutative 49.7%

      \[\leadsto \frac{0.16666666666666666}{\color{blue}{\beta + 2}} \]

10. Simplified 49.7%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

11. Final simplification 49.7%

    \[\leadsto \frac{0.16666666666666666}{\beta + 2} \]
12. Add Preprocessing

Alternative 21: 6.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 1.0 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 1.0 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 1.0 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(1.0 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 1.0 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(1.0 / beta), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Simplified 97.5%

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

5. Taylor expanded in beta around inf 28.5%

   \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]
6. Step-by-step derivation

   1. un-div-inv 28.6%

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

   2. +-commutative 28.6%

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

7. Applied egg-rr 28.6%

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

8. Taylor expanded in alpha around inf 4.3%

   \[\leadsto \frac{\color{blue}{1}}{\beta} \]

9. Final simplification 4.3%

   \[\leadsto \frac{1}{\beta} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))