Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 17.3s

Alternatives: 15

Speedup: 3.2×

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: 94.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.0%

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

   1. associate-/l/ 94.6%

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

   2. associate-/r* 85.2%

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

   3. +-commutative 85.2%

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

   4. associate-+r+ 85.2%

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

   5. +-commutative 85.2%

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

   6. associate-+r+ 85.2%

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

   7. associate-+r+ 85.2%

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

   8. distribute-rgt1-in 85.2%

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

   9. +-commutative 85.2%

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

   10. *-commutative 85.2%

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

   11. distribute-rgt1-in 85.2%

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

   12. +-commutative 85.2%

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

   13. metadata-eval 85.2%

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

   14. associate-+l+ 85.2%

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

   15. *-commutative 85.2%

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

   16. metadata-eval 85.2%

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

   17. associate-+l+ 85.2%

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

3. Simplified 85.2%

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

   1. *-commutative 85.2%

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

   2. frac-times 96.4%

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

   3. *-commutative 96.4%

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

   4. associate-+r+ 96.4%

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

   5. associate-/r* 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \beta}{t_1} \cdot \frac{\alpha + 1}{t_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_1}}{t_0} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

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))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+153)
     (* (/ (+ 1.0 beta) t_1) (/ (+ alpha 1.0) (* t_1 t_0)))
     (* (/ (/ (+ alpha 1.0) t_1) t_0) (+ 1.0 (/ (- -1.0 alpha) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+153) {
		tmp = ((1.0 + beta) / t_1) * ((alpha + 1.0) / (t_1 * t_0));
	} else {
		tmp = (((alpha + 1.0) / t_1) / t_0) * (1.0 + ((-1.0 - 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 5d+153) then
        tmp = ((1.0d0 + beta) / t_1) * ((alpha + 1.0d0) / (t_1 * t_0))
    else
        tmp = (((alpha + 1.0d0) / t_1) / t_0) * (1.0d0 + (((-1.0d0) - alpha) / beta))
    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 t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+153) {
		tmp = ((1.0 + beta) / t_1) * ((alpha + 1.0) / (t_1 * t_0));
	} else {
		tmp = (((alpha + 1.0) / t_1) / t_0) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+153:
		tmp = ((1.0 + beta) / t_1) * ((alpha + 1.0) / (t_1 * t_0))
	else:
		tmp = (((alpha + 1.0) / t_1) / t_0) * (1.0 + ((-1.0 - alpha) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+153)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_1) * Float64(Float64(alpha + 1.0) / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / t_1) / t_0) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+153)
		tmp = ((1.0 + beta) / t_1) * ((alpha + 1.0) / (t_1 * t_0));
	else
		tmp = (((alpha + 1.0) / t_1) / t_0) * (1.0 + ((-1.0 - 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+153], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000018e153

   1. Initial program 99.8%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 87.3%

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

      3. +-commutative 87.3%

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

      4. associate-+l+ 87.3%

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

      5. associate-+r+ 87.3%

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

      6. *-commutative 87.3%

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

      7. distribute-rgt1-in 87.3%

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

      8. +-commutative 87.3%

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

      9. *-commutative 87.3%

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

      10. distribute-rgt1-in 87.3%

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

      11. +-commutative 87.3%

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

      12. times-frac 98.6%

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

   3. Simplified 98.6%

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

   if 5.00000000000000018e153 < beta

   1. Initial program 77.6%

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

      1. associate-/l/ 75.1%

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

      2. associate-/r* 75.1%

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

      3. +-commutative 75.1%

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

      4. associate-+r+ 75.1%

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

      5. +-commutative 75.1%

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

      6. associate-+r+ 75.1%

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

      7. associate-+r+ 75.1%

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

      8. distribute-rgt1-in 75.1%

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

      9. +-commutative 75.1%

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

      10. *-commutative 75.1%

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

      11. distribute-rgt1-in 75.1%

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

      12. +-commutative 75.1%

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

      13. metadata-eval 75.1%

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

      14. associate-+l+ 75.1%

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

      15. *-commutative 75.1%

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

      16. metadata-eval 75.1%

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

      17. associate-+l+ 75.1%

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

   3. Simplified 75.1%

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

      1. *-commutative 75.1%

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

      2. frac-times 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-+r+ 86.2%

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

      5. associate-/r* 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in beta around inf 85.6%

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

      1. mul-1-neg 85.6%

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

   8. Simplified 85.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.95:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.95)
   (/ (/ (+ alpha 1.0) (+ alpha 2.0)) (* (+ alpha 2.0) (+ 3.0 (+ alpha beta))))
   (*
    (/ (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))) (+ alpha (+ beta 3.0)))
    (+ 1.0 (/ (- -1.0 alpha) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.95) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - 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 <= 3.95d0) then
        tmp = ((alpha + 1.0d0) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = (((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) / (alpha + (beta + 3.0d0))) * (1.0d0 + (((-1.0d0) - alpha) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.95) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.95:
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.95)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + Float64(beta + 3.0))) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.95)
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - 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, 3.95], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.9500000000000002

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-+l+ 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around 0 97.2%

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

   5. Taylor expanded in beta around 0 97.9%

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

   if 3.9500000000000002 < beta

   1. Initial program 87.9%

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

      1. associate-/l/ 84.5%

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

      2. associate-/r* 69.1%

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

      3. +-commutative 69.1%

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

      4. associate-+r+ 69.1%

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

      5. +-commutative 69.1%

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

      6. associate-+r+ 69.1%

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

      7. associate-+r+ 69.1%

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

      8. distribute-rgt1-in 69.1%

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

      9. +-commutative 69.1%

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

      10. *-commutative 69.1%

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

      11. distribute-rgt1-in 69.1%

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

      12. +-commutative 69.1%

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

      13. metadata-eval 69.1%

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

      14. associate-+l+ 69.1%

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

      15. *-commutative 69.1%

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

      16. metadata-eval 69.1%

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

      17. associate-+l+ 69.1%

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

   3. Simplified 69.1%

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

      1. *-commutative 69.1%

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

      2. frac-times 90.4%

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

      3. *-commutative 90.4%

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

      4. associate-+r+ 90.4%

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

      5. associate-/r* 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in beta around inf 88.2%

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

      1. mul-1-neg 88.2%

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

   8. Simplified 88.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.95:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. associate-/l/ 94.6%

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

   2. associate-/r* 85.2%

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

   3. +-commutative 85.2%

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

   4. associate-+l+ 85.2%

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

   5. associate-+r+ 85.2%

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

   6. *-commutative 85.2%

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

   7. distribute-rgt1-in 85.2%

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

   8. +-commutative 85.2%

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

   9. *-commutative 85.2%

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

   10. distribute-rgt1-in 85.2%

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

   11. +-commutative 85.2%

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

   12. times-frac 96.4%

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

3. Simplified 96.4%

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

   1. associate-*r/ 96.5%

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

   2. +-commutative 96.5%

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

5. Applied egg-rr 96.5%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

   9. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 5: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.3)
   (/ (/ (+ alpha 1.0) (+ alpha 2.0)) (* (+ alpha 2.0) (+ 3.0 (+ alpha beta))))
   (*
    (/ (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))) (+ alpha (+ beta 3.0)))
    (/ (+ 1.0 beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * ((1.0 + 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 <= 4.3d0) then
        tmp = ((alpha + 1.0d0) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = (((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) / (alpha + (beta + 3.0d0))) * ((1.0d0 + beta) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * ((1.0 + beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.3:
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * ((1.0 + beta) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.3)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 + beta) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.3)
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = (((alpha + 1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))) * ((1.0 + 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, 4.3], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.29999999999999982

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-+l+ 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around 0 97.2%

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

   5. Taylor expanded in beta around 0 97.9%

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

   if 4.29999999999999982 < beta

   1. Initial program 87.9%

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

      1. associate-/l/ 84.5%

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

      2. associate-/r* 69.1%

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

      3. +-commutative 69.1%

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

      4. associate-+r+ 69.1%

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

      5. +-commutative 69.1%

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

      6. associate-+r+ 69.1%

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

      7. associate-+r+ 69.1%

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

      8. distribute-rgt1-in 69.1%

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

      9. +-commutative 69.1%

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

      10. *-commutative 69.1%

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

      11. distribute-rgt1-in 69.1%

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

      12. +-commutative 69.1%

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

      13. metadata-eval 69.1%

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

      14. associate-+l+ 69.1%

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

      15. *-commutative 69.1%

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

      16. metadata-eval 69.1%

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

      17. associate-+l+ 69.1%

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

   3. Simplified 69.1%

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

      1. *-commutative 69.1%

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

      2. frac-times 90.4%

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

      3. *-commutative 90.4%

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

      4. associate-+r+ 90.4%

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

      5. associate-/r* 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in beta around inf 88.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\beta}\\ \end{array} \]

Alternative 6: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 33:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ alpha beta))))
   (if (<= beta 33.0)
     (/ (/ (+ alpha 1.0) (+ alpha 2.0)) (* (+ alpha 2.0) t_0))
     (/ (/ (+ alpha 1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 33.0) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (alpha + beta)
    if (beta <= 33.0d0) then
        tmp = ((alpha + 1.0d0) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * t_0)
    else
        tmp = ((alpha + 1.0d0) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 33.0) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (alpha + beta)
	tmp = 0
	if beta <= 33.0:
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * t_0)
	else:
		tmp = ((alpha + 1.0) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 33.0)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 33.0)
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + 2.0) * t_0);
	else
		tmp = ((alpha + 1.0) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 33.0], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 33

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-+l+ 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around 0 97.2%

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

   5. Taylor expanded in beta around 0 97.9%

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

   if 33 < beta

   1. Initial program 87.9%

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

   2. Taylor expanded in beta around inf 87.7%

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

   3. Taylor expanded in alpha around 0 87.7%

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

      1. +-commutative 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 33:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 7: 98.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8e+50)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8e+50) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (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 <= 8d+50) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8e+50) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8e+50:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8e+50)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8e+50)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (3.0 + (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, 8e+50], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.0000000000000006e50

   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/ 98.9%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. +-commutative 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-+l+ 98.9%

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

      7. metadata-eval 98.9%

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

      8. associate-+l+ 98.9%

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

      9. metadata-eval 98.9%

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

      10. +-commutative 98.9%

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

      11. metadata-eval 98.9%

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

      12. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

      1. associate-+r+ 98.9%

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

      2. +-commutative 98.9%

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

      3. flip-+ 98.9%

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

      4. pow2 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in alpha around inf 99.0%

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

   7. Taylor expanded in alpha around 0 62.8%

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

      1. +-commutative 62.8%

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

      2. *-commutative 62.8%

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

      3. associate--l+ 62.8%

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

      4. *-commutative 62.8%

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

      5. *-rgt-identity 62.8%

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

      6. distribute-lft-out-- 62.8%

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

      7. metadata-eval 62.8%

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

      8. *-rgt-identity 62.8%

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

      9. +-commutative 62.8%

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

   9. Simplified 62.8%

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

   if 8.0000000000000006e50 < beta

   1. Initial program 85.9%

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

   2. Taylor expanded in beta around inf 88.4%

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

   3. Taylor expanded in alpha around 0 88.4%

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

      1. +-commutative 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 8: 96.7% accurate, 2.7× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (+ alpha 1.0) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (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 <= 4.5d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((alpha + 1.0) / beta) / (3.0 + (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, 4.5], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-+l+ 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around 0 97.2%

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

   5. Taylor expanded in alpha around 0 59.9%

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

      1. associate-/r* 59.9%

         \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

      2. +-commutative 59.9%

         \[\leadsto \frac{\frac{0.5}{2 + \beta}}{\color{blue}{\beta + 3}} \]

   7. Simplified 59.9%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{\beta + 3}} \]

   if 4.5 < beta

   1. Initial program 87.9%

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

   2. Taylor expanded in beta around inf 87.7%

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

   3. Taylor expanded in alpha around 0 87.7%

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

      1. +-commutative 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 68.9%

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

Alternative 9: 91.5% accurate, 3.2× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ 0.5 (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) / (beta + 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 <= 4.5d0) then
        tmp = 0.5d0 / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(0.5 / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.5 / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = (1.0 / beta) / (beta + 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, 4.5], N[(0.5 / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-+l+ 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around 0 97.2%

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

   5. Taylor expanded in alpha around 0 59.9%

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

   if 4.5 < beta

   1. Initial program 87.9%

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

   2. Taylor expanded in beta around inf 87.7%

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

   3. Taylor expanded in alpha around 0 81.4%

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

      1. associate-/r* 81.4%

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

      2. +-commutative 81.4%

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

   5. Simplified 81.4%

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

4. Final simplification 66.9%

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

Alternative 10: 91.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.7:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.7)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.7) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta + 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 <= 4.7d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.7) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.7:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.7)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.7)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = (1.0 / beta) / (beta + 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, 4.7], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.70000000000000018

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-+l+ 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around 0 97.2%

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

   5. Taylor expanded in alpha around 0 59.9%

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

      1. associate-/r* 59.9%

         \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

      2. +-commutative 59.9%

         \[\leadsto \frac{\frac{0.5}{2 + \beta}}{\color{blue}{\beta + 3}} \]

   7. Simplified 59.9%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{\beta + 3}} \]

   if 4.70000000000000018 < beta

   1. Initial program 87.9%

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

   2. Taylor expanded in beta around inf 87.7%

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

   3. Taylor expanded in alpha around 0 81.4%

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

      1. associate-/r* 81.4%

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

      2. +-commutative 81.4%

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

   5. Simplified 81.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.7:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 11: 96.7% accurate, 3.2× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 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 <= 4.5d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((alpha + 1.0) / beta) / (beta + 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, 4.5], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-+l+ 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-+l+ 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around 0 97.2%

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

   5. Taylor expanded in alpha around 0 59.9%

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

      1. associate-/r* 59.9%

         \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{3 + \beta}} \]

      2. +-commutative 59.9%

         \[\leadsto \frac{\frac{0.5}{2 + \beta}}{\color{blue}{\beta + 3}} \]

   7. Simplified 59.9%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{2 + \beta}}{\beta + 3}} \]

   if 4.5 < beta

   1. Initial program 87.9%

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

   2. Taylor expanded in beta around inf 87.7%

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

   3. Taylor expanded in alpha around 0 87.6%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
   4. Step-by-step derivation

      1. +-commutative 87.6%

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

   5. Simplified 87.6%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 12: 49.7% 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 96.0%

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

2. Taylor expanded in beta around inf 30.4%

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

3. Taylor expanded in alpha around 0 28.4%

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

4. Final simplification 28.4%

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

Alternative 13: 50.1% 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 96.0%

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

2. Taylor expanded in beta around inf 30.4%

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

3. Taylor expanded in alpha around 0 28.4%

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

   1. associate-/r* 28.4%

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

   2. +-commutative 28.4%

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

5. Simplified 28.4%

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

6. Final simplification 28.4%

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

Alternative 14: 3.7% accurate, 11.7× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return -1.0 / beta;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. associate-/l/ 94.6%

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

   2. associate-/r* 85.2%

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

   3. +-commutative 85.2%

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

   4. associate-+l+ 85.2%

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

   5. associate-+r+ 85.2%

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

   6. *-commutative 85.2%

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

   7. distribute-rgt1-in 85.2%

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

   8. +-commutative 85.2%

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

   9. *-commutative 85.2%

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

   10. distribute-rgt1-in 85.2%

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

   11. +-commutative 85.2%

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

   12. times-frac 96.4%

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

3. Simplified 96.4%

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

4. Taylor expanded in alpha around inf 31.1%

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

5. Taylor expanded in beta around inf 2.6%

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

   1. associate-*r/ 2.6%

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

   2. distribute-lft-in 2.6%

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

   3. metadata-eval 2.6%

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

   4. mul-1-neg 2.6%

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

   5. unsub-neg 2.6%

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

7. Simplified 2.6%

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

8. Taylor expanded in alpha around inf 3.6%

   \[\leadsto \color{blue}{\frac{-1}{\beta}} \]

9. Final simplification 3.6%

   \[\leadsto \frac{-1}{\beta} \]

Alternative 15: 6.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / beta;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

2. Taylor expanded in beta around inf 30.4%

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

3. Taylor expanded in alpha around inf 4.3%

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

4. Final simplification 4.3%

   \[\leadsto \frac{1}{\beta} \]

Reproduce

herbie shell --seed 2023307 
(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)))