Octave 3.8, jcobi/3

Percentage Accurate: 94.0% → 99.8%

Time: 19.8s

Alternatives: 19

Speedup: 2.5×

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: 94.0% 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: 99.8% accurate, 1.4× speedup

Math

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

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 beta) (+ beta (+ alpha 3.0))) (/ (+ 1.0 alpha) t_0)) t_0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return (((1.0 + beta) / (beta + (alpha + 3.0))) * ((1.0 + alpha) / t_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 + beta) / (beta + (alpha + 3.0d0))) * ((1.0d0 + alpha) / t_0)) / 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 + beta) / (beta + (alpha + 3.0))) * ((1.0 + alpha) / t_0)) / t_0;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	return (((1.0 + beta) / (beta + (alpha + 3.0))) * ((1.0 + alpha) / t_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(Float64(1.0 + beta) / Float64(beta + Float64(alpha + 3.0))) * Float64(Float64(1.0 + alpha) / t_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 + beta) / (beta + (alpha + 3.0))) * ((1.0 + alpha) / t_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[(N[(1.0 + beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 94.1%

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

3. Simplified 85.5%

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

5. Taylor expanded in alpha around 0 85.6%

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

   1. times-frac 97.3%

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

   2. +-commutative 97.3%

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

   3. +-commutative 97.3%

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

   4. associate-+r+ 97.3%

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

   5. +-commutative 97.3%

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

   6. associate-+l+ 97.3%

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

   7. +-commutative 97.3%

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

   8. +-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-+l+ 97.3%

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

   11. +-commutative 97.3%

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

   12. associate-+l+ 97.3%

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

7. Applied egg-rr 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

9. Simplified 99.8%

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

    1. associate-*l/ 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

    5. +-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.45e+62)
   (/
    (* (+ 1.0 beta) (+ 1.0 alpha))
    (* (+ alpha (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (/ (- alpha -1.0) (+ alpha (+ beta 3.0))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45e+62) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	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 <= 2.45d+62) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / ((alpha + (beta + 2.0d0)) * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (beta + 3.0d0))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45e+62) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.45e+62:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.45e+62)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.45e+62)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	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, 2.45e+62], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $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[(alpha - -1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4499999999999998e62

   1. Initial program 99.3%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in alpha around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

   7. Simplified 68.5%

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

   if 2.4499999999999998e62 < beta

   1. Initial program 80.5%

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

      1. associate-/l/ 76.2%

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

      2. +-commutative 76.2%

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

      3. associate-+l+ 76.2%

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

      4. *-commutative 76.2%

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

      5. metadata-eval 76.2%

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

      6. associate-+l+ 76.2%

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

      7. metadata-eval 76.2%

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

      8. +-commutative 76.2%

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

      9. +-commutative 76.2%

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

      10. +-commutative 76.2%

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

      11. metadata-eval 76.2%

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

      12. metadata-eval 76.2%

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

      13. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in beta around -inf 86.0%

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

      1. mul-1-neg 86.0%

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

      2. sub-neg 86.0%

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

      3. mul-1-neg 86.0%

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

      4. distribute-neg-in 86.0%

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

      5. +-commutative 86.0%

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

      6. mul-1-neg 86.0%

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

      7. distribute-lft-in 86.0%

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

      8. metadata-eval 86.0%

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

      9. mul-1-neg 86.0%

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

      10. unsub-neg 86.0%

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

   7. Simplified 86.0%

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

      1. *-un-lft-identity 86.0%

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

      2. associate-+l+ 86.0%

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

      3. associate-+r+ 86.0%

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

      4. +-commutative 86.0%

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

      5. associate-+r+ 86.0%

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

   9. Applied egg-rr 86.0%

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

       1. *-lft-identity 86.0%

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

       2. associate-/r* 86.8%

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

       3. +-commutative 86.8%

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

       4. +-commutative 86.8%

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

       5. +-commutative 86.8%

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

       6. associate-+r+ 86.8%

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

   11. Simplified 86.8%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8e+62)
   (/
    (* (+ 1.0 beta) (+ 1.0 alpha))
    (* (+ alpha (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0))))
   (*
    (/ (+ 1.0 alpha) (+ beta (+ alpha 2.0)))
    (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8e+62) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / 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 <= 1.8d+62) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / ((alpha + (beta + 2.0d0)) * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 2.0d0))) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8e+62) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.8e+62:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8e+62)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / 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(beta + Float64(alpha + 2.0))) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.8e+62)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / 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, 1.8e+62], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8e62

   1. Initial program 99.3%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in alpha around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

   7. Simplified 68.5%

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

   if 1.8e62 < beta

   1. Initial program 80.5%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in alpha around 0 61.6%

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

      1. times-frac 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+r+ 91.1%

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

      5. +-commutative 91.1%

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

      6. associate-+l+ 91.1%

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

      7. +-commutative 91.1%

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

      8. +-commutative 91.1%

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

      9. *-commutative 91.1%

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

      10. associate-+l+ 91.1%

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

      11. +-commutative 91.1%

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

      12. associate-+l+ 91.1%

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

   7. Applied egg-rr 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in beta around inf 86.1%

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

       1. mul-1-neg 86.1%

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

       2. *-commutative 86.1%

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

   12. Simplified 86.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

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 beta) (+ beta (+ alpha 3.0))) t_0) (/ (+ 1.0 alpha) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return (((1.0 + beta) / (beta + (alpha + 3.0))) / t_0) * ((1.0 + alpha) / 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 + beta) / (beta + (alpha + 3.0d0))) / t_0) * ((1.0d0 + alpha) / 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 + beta) / (beta + (alpha + 3.0))) / t_0) * ((1.0 + alpha) / t_0);
}

Python

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = (((1.0 + beta) / (beta + (alpha + 3.0))) / t_0) * ((1.0 + alpha) / 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[(N[(1.0 + beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.1%

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

3. Simplified 85.5%

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

5. Taylor expanded in alpha around 0 85.6%

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

   1. times-frac 97.3%

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

   2. +-commutative 97.3%

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

   3. +-commutative 97.3%

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

   4. associate-+r+ 97.3%

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

   5. +-commutative 97.3%

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

   6. associate-+l+ 97.3%

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

   7. +-commutative 97.3%

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

   8. +-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-+l+ 97.3%

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

   11. +-commutative 97.3%

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

   12. associate-+l+ 97.3%

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

7. Applied egg-rr 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

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 3.0))))
   (if (<= beta 2e+37)
     (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (* (+ beta 2.0) t_0)))
     (/ (/ (- alpha -1.0) t_0) (+ 2.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2e+37) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * t_0));
	} else {
		tmp = ((alpha - -1.0) / t_0) / (2.0 + (beta + alpha));
	}
	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 + 3.0d0)
    if (beta <= 2d+37) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * ((beta + 2.0d0) * t_0))
    else
        tmp = ((alpha - (-1.0d0)) / t_0) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2e+37) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * t_0));
	} else {
		tmp = ((alpha - -1.0) / t_0) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 2e+37:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * t_0))
	else:
		tmp = ((alpha - -1.0) / t_0) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 2e+37)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 2e+37)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * t_0));
	else
		tmp = ((alpha - -1.0) / t_0) / (2.0 + (beta + alpha));
	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 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+37], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999991e37

   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 96.2%

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

   5. Taylor expanded in alpha around 0 85.6%

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

      1. +-commutative 85.6%

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

   7. Simplified 85.6%

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

   8. Taylor expanded in alpha around 0 87.3%

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

   if 1.99999999999999991e37 < beta

   1. Initial program 80.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. associate-+l+ 76.4%

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

      4. *-commutative 76.4%

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

      5. metadata-eval 76.4%

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

      6. associate-+l+ 76.4%

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

      7. metadata-eval 76.4%

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

      8. +-commutative 76.4%

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

      9. +-commutative 76.4%

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

      10. +-commutative 76.4%

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

      11. metadata-eval 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in beta around -inf 83.2%

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

      1. mul-1-neg 83.2%

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

      2. sub-neg 83.2%

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

      3. mul-1-neg 83.2%

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

      4. distribute-neg-in 83.2%

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

      5. +-commutative 83.2%

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

      6. mul-1-neg 83.2%

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

      7. distribute-lft-in 83.2%

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

      8. metadata-eval 83.2%

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

      9. mul-1-neg 83.2%

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

      10. unsub-neg 83.2%

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

   7. Simplified 83.2%

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

      1. *-un-lft-identity 83.2%

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

      2. associate-+l+ 83.2%

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

      3. associate-+r+ 83.2%

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

      4. +-commutative 83.2%

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

      5. associate-+r+ 83.2%

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

   9. Applied egg-rr 83.2%

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

       1. *-lft-identity 83.2%

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

       2. associate-/r* 82.9%

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

       3. +-commutative 82.9%

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

       4. +-commutative 82.9%

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

       5. +-commutative 82.9%

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

       6. associate-+r+ 82.9%

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

   11. Simplified 82.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.06e+32)
   (/
    (/ (+ 1.0 beta) (+ beta 2.0))
    (* (+ alpha (+ beta 2.0)) (+ 3.0 (+ beta alpha))))
   (/ (/ (- alpha -1.0) (+ alpha (+ beta 3.0))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.06e+32) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	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 <= 1.06d+32) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((alpha + (beta + 2.0d0)) * (3.0d0 + (beta + alpha)))
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (beta + 3.0d0))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.06e+32) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.06e+32:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (3.0 + (beta + alpha)))
	else:
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.06e+32)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.06e+32)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (3.0 + (beta + alpha)));
	else
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	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, 1.06e+32], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.0600000000000001e32

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 87.1%

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

      1. +-commutative 87.1%

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

   7. Simplified 87.1%

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

   if 1.0600000000000001e32 < beta

   1. Initial program 80.9%

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

      1. associate-/l/ 77.0%

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

      2. +-commutative 77.0%

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

      3. associate-+l+ 77.0%

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

      4. *-commutative 77.0%

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

      5. metadata-eval 77.0%

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

      6. associate-+l+ 77.0%

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

      7. metadata-eval 77.0%

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

      8. +-commutative 77.0%

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

      9. +-commutative 77.0%

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

      10. +-commutative 77.0%

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

      11. metadata-eval 77.0%

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

      12. metadata-eval 77.0%

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

      13. associate-+l+ 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in beta around -inf 83.6%

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

      1. mul-1-neg 83.6%

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

      2. sub-neg 83.6%

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

      3. mul-1-neg 83.6%

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

      4. distribute-neg-in 83.6%

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

      5. +-commutative 83.6%

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

      6. mul-1-neg 83.6%

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

      7. distribute-lft-in 83.6%

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

      8. metadata-eval 83.6%

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

      9. mul-1-neg 83.6%

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

      10. unsub-neg 83.6%

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

   7. Simplified 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. associate-+l+ 83.6%

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

      3. associate-+r+ 83.6%

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

      4. +-commutative 83.6%

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

      5. associate-+r+ 83.6%

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

   9. Applied egg-rr 83.6%

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

       1. *-lft-identity 83.6%

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

       2. associate-/r* 83.3%

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

       3. +-commutative 83.3%

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

       4. +-commutative 83.3%

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

       5. +-commutative 83.3%

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

       6. associate-+r+ 83.3%

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

   11. Simplified 83.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.5× speedup

Math

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

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) t_0) (/ (/ (+ 1.0 beta) (+ 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) / t_0) * (((1.0 + beta) / (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) / t_0) * (((1.0d0 + beta) / (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) / t_0) * (((1.0 + beta) / (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) / t_0) * (((1.0 + beta) / (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) / t_0) * Float64(Float64(Float64(1.0 + beta) / 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) / t_0) * (((1.0 + beta) / (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] / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.1%

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

3. Simplified 85.5%

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

5. Taylor expanded in alpha around 0 85.6%

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

   1. times-frac 97.3%

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

   2. +-commutative 97.3%

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

   3. +-commutative 97.3%

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

   4. associate-+r+ 97.3%

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

   5. +-commutative 97.3%

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

   6. associate-+l+ 97.3%

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

   7. +-commutative 97.3%

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

   8. +-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-+l+ 97.3%

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

   11. +-commutative 97.3%

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

   12. associate-+l+ 97.3%

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

7. Applied egg-rr 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in alpha around 0 73.9%

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

11. Final simplification 73.9%

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

Alternative 8: 98.5% accurate, 1.5× speedup

Math

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

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) t_0) (/ (+ 1.0 beta) (+ 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) / t_0) * ((1.0 + beta) / (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) / t_0) * ((1.0d0 + beta) / (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) / t_0) * ((1.0 + beta) / (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) / t_0) * ((1.0 + beta) / (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(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + beta) / 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) / t_0) * ((1.0 + beta) / (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[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 94.1%

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

3. Simplified 85.5%

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

5. Taylor expanded in alpha around 0 85.6%

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

   1. times-frac 97.3%

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

   2. +-commutative 97.3%

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

   3. +-commutative 97.3%

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

   4. associate-+r+ 97.3%

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

   5. +-commutative 97.3%

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

   6. associate-+l+ 97.3%

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

   7. +-commutative 97.3%

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

   8. +-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-+l+ 97.3%

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

   11. +-commutative 97.3%

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

   12. associate-+l+ 97.3%

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

7. Applied egg-rr 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

9. Simplified 99.8%

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

    1. associate-*l/ 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

    5. +-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

12. Taylor expanded in alpha around 0 73.9%

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

13. Final simplification 73.9%

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

Alternative 9: 97.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.05)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ 3.0 (+ beta alpha))))
   (/ (/ (- alpha -1.0) (+ alpha (+ beta 3.0))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	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 <= 2.05d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (3.0d0 + (beta + alpha)))
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (beta + 3.0d0))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.05:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)))
	else:
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.05)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.05)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)));
	else
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	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, 2.05], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0499999999999998

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in beta around 0 98.7%

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

   if 2.0499999999999998 < beta

   1. Initial program 82.4%

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

      1. associate-/l/ 78.8%

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

      2. +-commutative 78.8%

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

      3. associate-+l+ 78.8%

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

      4. *-commutative 78.8%

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

      5. metadata-eval 78.8%

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

      6. associate-+l+ 78.8%

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

      7. metadata-eval 78.8%

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

      8. +-commutative 78.8%

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

      9. +-commutative 78.8%

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

      10. +-commutative 78.8%

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

      11. metadata-eval 78.8%

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

      12. metadata-eval 78.8%

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

      13. associate-+l+ 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in beta around -inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. sub-neg 81.2%

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

      3. mul-1-neg 81.2%

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

      4. distribute-neg-in 81.2%

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

      5. +-commutative 81.2%

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

      6. mul-1-neg 81.2%

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

      7. distribute-lft-in 81.2%

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

      8. metadata-eval 81.2%

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

      9. mul-1-neg 81.2%

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

      10. unsub-neg 81.2%

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

   7. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. associate-+l+ 81.2%

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

      3. associate-+r+ 81.2%

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

      4. +-commutative 81.2%

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

      5. associate-+r+ 81.2%

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

   9. Applied egg-rr 81.2%

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

       1. *-lft-identity 81.2%

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

       2. associate-/r* 79.8%

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

       3. +-commutative 79.8%

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

       4. +-commutative 79.8%

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

       5. +-commutative 79.8%

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

       6. associate-+r+ 79.8%

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

   11. Simplified 79.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 3.0) (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (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 <= 6.0d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 3.0d0) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (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 <= 6.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (3.0 + (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, 6.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 97.7%

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

   6. Taylor expanded in beta around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. +-commutative 97.7%

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

   8. Simplified 97.7%

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

   if 6 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around inf 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. associate-+l+ 79.9%

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

      4. metadata-eval 79.9%

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

      5. associate-+r+ 79.9%

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

      6. +-commutative 79.9%

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

      7. associate-+l+ 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. associate-*l/ 79.9%

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

      2. +-commutative 79.9%

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

   7. Applied egg-rr 79.9%

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

      1. associate-*r/ 79.9%

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

      2. *-rgt-identity 79.9%

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

      3. +-commutative 79.9%

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

      4. +-commutative 79.9%

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

      5. associate-+r+ 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 3.0) (+ alpha 2.0)))
   (/ (/ (- alpha -1.0) (+ alpha (+ beta 3.0))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	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 <= 1.8d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 3.0d0) * (alpha + 2.0d0))
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (beta + 3.0d0))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.8:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0))
	else:
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.8)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	else
		tmp = ((alpha - -1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	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, 1.8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.80000000000000004

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in beta around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. +-commutative 98.2%

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

   8. Simplified 98.2%

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

   if 1.80000000000000004 < beta

   1. Initial program 82.4%

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

      1. associate-/l/ 78.8%

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

      2. +-commutative 78.8%

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

      3. associate-+l+ 78.8%

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

      4. *-commutative 78.8%

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

      5. metadata-eval 78.8%

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

      6. associate-+l+ 78.8%

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

      7. metadata-eval 78.8%

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

      8. +-commutative 78.8%

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

      9. +-commutative 78.8%

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

      10. +-commutative 78.8%

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

      11. metadata-eval 78.8%

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

      12. metadata-eval 78.8%

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

      13. associate-+l+ 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in beta around -inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. sub-neg 81.2%

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

      3. mul-1-neg 81.2%

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

      4. distribute-neg-in 81.2%

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

      5. +-commutative 81.2%

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

      6. mul-1-neg 81.2%

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

      7. distribute-lft-in 81.2%

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

      8. metadata-eval 81.2%

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

      9. mul-1-neg 81.2%

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

      10. unsub-neg 81.2%

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

   7. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. associate-+l+ 81.2%

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

      3. associate-+r+ 81.2%

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

      4. +-commutative 81.2%

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

      5. associate-+r+ 81.2%

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

   9. Applied egg-rr 81.2%

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

       1. *-lft-identity 81.2%

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

       2. associate-/r* 79.8%

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

       3. +-commutative 79.8%

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

       4. +-commutative 79.8%

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

       5. +-commutative 79.8%

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

       6. associate-+r+ 79.8%

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

   11. Simplified 79.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 19:\\ \;\;\;\;0.5 \cdot \frac{1 + \alpha}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 19.0)
   (* 0.5 (/ (+ 1.0 alpha) (* (+ alpha 3.0) (+ alpha 2.0))))
   (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 19.0) {
		tmp = 0.5 * ((1.0 + alpha) / ((alpha + 3.0) * (alpha + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (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 <= 19.0d0) then
        tmp = 0.5d0 * ((1.0d0 + alpha) / ((alpha + 3.0d0) * (alpha + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (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 <= 19.0) {
		tmp = 0.5 * ((1.0 + alpha) / ((alpha + 3.0) * (alpha + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 19.0:
		tmp = 0.5 * ((1.0 + alpha) / ((alpha + 3.0) * (alpha + 2.0)))
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 19.0)
		tmp = Float64(0.5 * Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 19.0)
		tmp = 0.5 * ((1.0 + alpha) / ((alpha + 3.0) * (alpha + 2.0)));
	else
		tmp = ((1.0 + alpha) / (3.0 + (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, 19.0], N[(0.5 * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 19

   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 96.6%

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

   5. Taylor expanded in alpha around 0 86.0%

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

      1. +-commutative 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in beta around 0 84.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 84.7%

         \[\leadsto 0.5 \cdot \frac{1 + \alpha}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

      2. +-commutative 84.7%

         \[\leadsto 0.5 \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   10. Simplified 84.7%

       \[\leadsto \color{blue}{0.5 \cdot \frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   if 19 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around inf 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. associate-+l+ 79.9%

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

      4. metadata-eval 79.9%

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

      5. associate-+r+ 79.9%

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

      6. +-commutative 79.9%

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

      7. associate-+l+ 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. associate-*l/ 79.9%

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

      2. +-commutative 79.9%

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

   7. Applied egg-rr 79.9%

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

      1. associate-*r/ 79.9%

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

      2. *-rgt-identity 79.9%

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

      3. +-commutative 79.9%

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

      4. +-commutative 79.9%

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

      5. associate-+r+ 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 19:\\ \;\;\;\;0.5 \cdot \frac{1 + \alpha}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (/ 0.5 (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	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 <= 4.4d0) then
        tmp = 0.5d0 / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.4:
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.4)
		tmp = Float64(0.5 / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4)
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	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, 4.4], N[(0.5 / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 97.7%

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

   6. Taylor expanded in alpha around 0 68.2%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 4.4000000000000004 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around inf 79.9%

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

      1. *-un-lft-identity 79.9%

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

      2. metadata-eval 79.9%

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

      3. associate-+l+ 79.9%

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

      4. metadata-eval 79.9%

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

      5. associate-+r+ 79.9%

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

      6. +-commutative 79.9%

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

      7. associate-+l+ 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. *-lft-identity 79.9%

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

      2. +-commutative 79.9%

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

   7. Simplified 79.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (/ 0.5 (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (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 <= 4.4d0) then
        tmp = 0.5d0 / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (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 <= 4.4) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.4:
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.4)
		tmp = Float64(0.5 / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4)
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / (3.0 + (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, 4.4], N[(0.5 / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 97.7%

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

   6. Taylor expanded in alpha around 0 68.2%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 4.4000000000000004 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around inf 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. associate-+l+ 79.9%

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

      4. metadata-eval 79.9%

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

      5. associate-+r+ 79.9%

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

      6. +-commutative 79.9%

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

      7. associate-+l+ 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. associate-*l/ 79.9%

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

      2. +-commutative 79.9%

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

   7. Applied egg-rr 79.9%

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

      1. associate-*r/ 79.9%

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

      2. *-rgt-identity 79.9%

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

      3. +-commutative 79.9%

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

      4. +-commutative 79.9%

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

      5. associate-+r+ 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.2)
   (/ 0.5 (* (+ beta 3.0) (+ beta 2.0)))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / 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 <= 9.2d0) then
        tmp = 0.5d0 / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.2:
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.2)
		tmp = Float64(0.5 / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.2)
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / 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, 9.2], N[(0.5 / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.1999999999999993

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 97.7%

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

   6. Taylor expanded in alpha around 0 68.2%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 9.1999999999999993 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around inf 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. associate-+l+ 79.9%

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

      4. metadata-eval 79.9%

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

      5. associate-+r+ 79.9%

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

      6. +-commutative 79.9%

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

      7. associate-+l+ 79.9%

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

   5. Applied egg-rr 79.9%

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

   6. Taylor expanded in beta around inf 79.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((1.0 + alpha) / beta) * (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 + alpha) / beta) * (1.0d0 / beta)
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
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] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 28.3%

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

   1. div-inv 28.2%

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

   2. metadata-eval 28.2%

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

   3. associate-+l+ 28.2%

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

   4. metadata-eval 28.2%

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

   5. associate-+r+ 28.2%

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

   6. +-commutative 28.2%

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

   7. associate-+l+ 28.2%

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

5. Applied egg-rr 28.2%

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

6. Taylor expanded in beta around inf 28.4%

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

7. Final simplification 28.4%

   \[\leadsto \frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta} \]
8. Add Preprocessing

Alternative 17: 50.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / (beta * (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
    code = 1.0d0 / (beta * (beta + 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 28.3%

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

4. Taylor expanded in alpha around 0 26.4%

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

   1. +-commutative 26.4%

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

6. Simplified 26.4%

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

7. Final simplification 26.4%

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

Alternative 18: 50.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 / beta) / (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
    code = (1.0d0 / beta) / (beta + 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 28.3%

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

   1. div-inv 28.2%

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

   2. metadata-eval 28.2%

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

   3. associate-+l+ 28.2%

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

   4. metadata-eval 28.2%

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

   5. associate-+r+ 28.2%

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

   6. +-commutative 28.2%

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

   7. associate-+l+ 28.2%

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

5. Applied egg-rr 28.2%

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

6. Taylor expanded in alpha around 0 26.4%

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

   1. associate-/r* 26.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{3 + \beta}} \]

8. Simplified 26.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{3 + \beta}} \]

9. Final simplification 26.4%

   \[\leadsto \frac{\frac{1}{\beta}}{\beta + 3} \]
10. Add Preprocessing

Alternative 19: 6.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 / 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 = 0.3333333333333333d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around inf 28.3%

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

4. Taylor expanded in alpha around 0 26.4%

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

   1. +-commutative 26.4%

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

6. Simplified 26.4%

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

7. Taylor expanded in beta around 0 4.2%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]

8. Final simplification 4.2%

   \[\leadsto \frac{0.3333333333333333}{\beta} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))