Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 20.3s

Alternatives: 19

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

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (t_0 / (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
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = ((1.0d0 + alpha) / t_0) / ((alpha + (beta + 3.0d0)) * (t_0 / (1.0d0 + beta)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.2%

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

3. Simplified 97.2%

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

   1. *-commutative 97.2%

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

   2. clear-num 97.2%

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

   3. clear-num 97.1%

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

   4. frac-times 97.2%

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

   5. metadata-eval 97.2%

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

   6. +-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-/r* 97.2%

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

   3. +-commutative 97.2%

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

   4. associate-/l* 99.3%

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

   5. +-commutative 99.3%

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

   6. +-commutative 99.3%

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

   7. +-commutative 99.3%

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

7. Simplified 99.3%

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

   1. div-inv 99.3%

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

   2. +-commutative 99.3%

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

   3. clear-num 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-/r/ 99.3%

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

   6. +-commutative 99.3%

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

9. Applied egg-rr 99.3%

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

    1. associate-*r/ 99.4%

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

    2. *-rgt-identity 99.4%

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

    3. +-commutative 99.4%

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

    4. +-commutative 99.4%

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

    5. *-commutative 99.4%

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

    6. +-commutative 99.4%

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

    7. +-commutative 99.4%

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

    8. +-commutative 99.4%

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

    9. +-commutative 99.4%

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

    10. +-commutative 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 14500000:\\ \;\;\;\;\frac{1 + \beta}{t_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(1 + \frac{\alpha - -1}{\beta}\right)}\\ \end{array} \end{array} \]

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 2.0))))
   (if (<= beta 14500000.0)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/
      (/ (+ 1.0 alpha) t_0)
      (* (+ alpha (+ beta 3.0)) (+ 1.0 (/ (- alpha -1.0) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 14500000.0) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 14500000.0d0) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / ((alpha + (beta + 3.0d0)) * (1.0d0 + ((alpha - (-1.0d0)) / beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 14500000.0) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 14500000.0:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 14500000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(1.0 + Float64(Float64(alpha - -1.0) / beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 14500000.0)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 14500000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.45e7

   1. Initial program 99.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in alpha around 0 81.1%

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

   5. Taylor expanded in alpha around 0 67.5%

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

   if 1.45e7 < beta

   1. Initial program 85.4%

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

   3. Simplified 92.4%

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

      1. *-commutative 92.4%

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

      2. clear-num 92.3%

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

      3. clear-num 92.3%

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

      4. frac-times 92.5%

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

      5. metadata-eval 92.5%

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

      6. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 92.4%

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

      3. +-commutative 92.4%

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

      4. associate-/l* 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. div-inv 99.0%

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

      2. +-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-/r/ 99.0%

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

      6. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.1%

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

       2. *-rgt-identity 99.1%

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

       3. +-commutative 99.1%

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

       4. +-commutative 99.1%

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

       5. *-commutative 99.1%

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

       6. +-commutative 99.1%

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

       7. +-commutative 99.1%

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

       8. +-commutative 99.1%

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

       9. +-commutative 99.1%

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

       10. +-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in beta around -inf 99.1%

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

       1. mul-1-neg 99.1%

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

       2. unsub-neg 99.1%

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

       3. mul-1-neg 99.1%

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

       4. distribute-neg-in 99.1%

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

       5. metadata-eval 99.1%

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

       6. associate-+r+ 99.1%

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

       7. metadata-eval 99.1%

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

       8. sub-neg 99.1%

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

   14. Simplified 99.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 14500000:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(1 + \frac{\alpha - -1}{\beta}\right)}\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 24000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{t_0}}{t_0 \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(1 + \frac{\alpha - -1}{\beta}\right)}\\ \end{array} \end{array} \]

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 2.0))))
   (if (<= beta 24000000.0)
     (/ (/ (+ 1.0 (+ alpha beta)) t_0) (* t_0 (+ 3.0 (+ alpha beta))))
     (/
      (/ (+ 1.0 alpha) t_0)
      (* (+ alpha (+ beta 3.0)) (+ 1.0 (/ (- alpha -1.0) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 24000000.0) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 24000000.0d0) then
        tmp = ((1.0d0 + (alpha + beta)) / t_0) / (t_0 * (3.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / ((alpha + (beta + 3.0d0)) * (1.0d0 + ((alpha - (-1.0d0)) / beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 24000000.0) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 24000000.0:
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 24000000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + beta)) / t_0) / Float64(t_0 * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(1.0 + Float64(Float64(alpha - -1.0) / beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 24000000.0)
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * (1.0 + ((alpha - -1.0) / beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 24000000.0], N[(N[(N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4e7

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. *-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+l+ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around 0 99.0%

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

   if 2.4e7 < beta

   1. Initial program 85.4%

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

   3. Simplified 92.4%

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

      1. *-commutative 92.4%

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

      2. clear-num 92.3%

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

      3. clear-num 92.3%

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

      4. frac-times 92.5%

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

      5. metadata-eval 92.5%

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

      6. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 92.4%

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

      3. +-commutative 92.4%

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

      4. associate-/l* 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. div-inv 99.0%

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

      2. +-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-/r/ 99.0%

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

      6. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.1%

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

       2. *-rgt-identity 99.1%

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

       3. +-commutative 99.1%

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

       4. +-commutative 99.1%

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

       5. *-commutative 99.1%

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

       6. +-commutative 99.1%

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

       7. +-commutative 99.1%

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

       8. +-commutative 99.1%

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

       9. +-commutative 99.1%

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

       10. +-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in beta around -inf 99.1%

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

       1. mul-1-neg 99.1%

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

       2. unsub-neg 99.1%

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

       3. mul-1-neg 99.1%

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

       4. distribute-neg-in 99.1%

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

       5. metadata-eval 99.1%

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

       6. associate-+r+ 99.1%

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

       7. metadata-eval 99.1%

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

       8. sub-neg 99.1%

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

   14. Simplified 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 24000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(1 + \frac{\alpha - -1}{\beta}\right)}\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

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 alpha) t_0) (/ (/ (+ 1.0 beta) t_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 + alpha) / t_0) * (((1.0 + beta) / t_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 + alpha) / t_0) * (((1.0d0 + beta) / t_0) / (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 + alpha) / t_0) * (((1.0 + beta) / t_0) / (alpha + (beta + 3.0)));
}

Python

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

Derivation

1. Initial program 95.2%

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

3. Simplified 97.2%

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

   1. clear-num 97.2%

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

   2. associate-+r+ 97.2%

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

   3. *-commutative 97.2%

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

   4. frac-times 95.1%

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

   5. *-un-lft-identity 95.1%

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

   6. +-commutative 95.1%

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

   7. *-commutative 95.1%

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

   8. associate-+r+ 95.1%

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

5. Applied egg-rr 95.1%

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

   1. associate-/r* 97.2%

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

   2. associate-/l* 94.2%

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

   3. associate-*l/ 97.2%

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

   4. *-commutative 97.2%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{t_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

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))))
   (if (<= beta 8.4e+15)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (- alpha -1.0) t_0) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 8.4e+15) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / t_0) / (alpha + (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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 8.4d+15) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / t_0) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 8.4e+15) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / t_0) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 8.4e+15:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((alpha - -1.0) / t_0) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 8.4e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 8.4e+15)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha - -1.0) / t_0) / (alpha + (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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8.4e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.4e15

   1. Initial program 99.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in alpha around 0 81.3%

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

   5. Taylor expanded in alpha around 0 67.9%

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

   if 8.4e15 < beta

   1. Initial program 85.1%

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

      1. associate-/l/ 82.6%

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

      2. +-commutative 82.6%

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

      3. +-commutative 82.6%

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

      4. associate-+r+ 82.6%

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

      5. *-commutative 82.6%

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

      6. metadata-eval 82.6%

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

      7. associate-+l+ 82.6%

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

      8. metadata-eval 82.6%

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

      9. associate-+l+ 82.6%

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

      10. metadata-eval 82.6%

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

      11. metadata-eval 82.6%

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

      12. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in beta around -inf 86.6%

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

      1. expm1-log1p-u 86.6%

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

      2. expm1-udef 51.0%

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

      3. mul-1-neg 51.0%

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

      4. *-commutative 51.0%

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

      5. fma-neg 51.0%

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

      6. metadata-eval 51.0%

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

      7. *-commutative 51.0%

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

      8. associate-+r+ 51.0%

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

      9. +-commutative 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. expm1-def 86.6%

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

      2. expm1-log1p 86.6%

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

      3. associate-/r* 84.1%

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

   8. Simplified 84.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 56000000:\\ \;\;\;\;\frac{1 + \beta}{t_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 56000000.0)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) t_0) (+ (+ beta 4.0) (* alpha 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 56000000.0) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 56000000.0d0) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / ((beta + 4.0d0) + (alpha * 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 56000000.0) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 56000000.0:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) / ((beta + 4.0) + (alpha * 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 56000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 56000000.0)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) / ((beta + 4.0) + (alpha * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 56000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.6e7

   1. Initial program 99.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in alpha around 0 81.1%

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

   5. Taylor expanded in alpha around 0 67.5%

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

   if 5.6e7 < beta

   1. Initial program 85.4%

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

   3. Simplified 92.4%

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

      1. *-commutative 92.4%

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

      2. clear-num 92.3%

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

      3. clear-num 92.3%

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

      4. frac-times 92.5%

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

      5. metadata-eval 92.5%

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

      6. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 92.4%

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

      3. +-commutative 92.4%

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

      4. associate-/l* 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. div-inv 99.0%

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

      2. +-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-/r/ 99.0%

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

      6. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.1%

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

       2. *-rgt-identity 99.1%

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

       3. +-commutative 99.1%

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

       4. +-commutative 99.1%

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

       5. *-commutative 99.1%

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

       6. +-commutative 99.1%

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

       7. +-commutative 99.1%

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

       8. +-commutative 99.1%

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

       9. +-commutative 99.1%

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

       10. +-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in beta around inf 84.6%

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

       1. associate-+r+ 84.6%

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

   14. Simplified 84.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 56000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

Alternative 7: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 160000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 160000000.0)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ beta 4.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 160000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0);
	}
	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 <= 160000000.0d0) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (beta + 4.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 160000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 160000000.0:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 160000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(beta + 4.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 160000000.0)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0);
	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, 160000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6e8

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. associate-*r/ 99.5%

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

      2. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in alpha around 0 83.0%

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

      1. +-commutative 83.0%

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

   8. Simplified 83.0%

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

   9. Taylor expanded in alpha around 0 66.6%

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

   if 1.6e8 < beta

   1. Initial program 85.4%

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

   3. Simplified 92.4%

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

      1. *-commutative 92.4%

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

      2. clear-num 92.3%

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

      3. clear-num 92.3%

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

      4. frac-times 92.5%

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

      5. metadata-eval 92.5%

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

      6. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 92.4%

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

      3. +-commutative 92.4%

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

      4. associate-/l* 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. div-inv 99.0%

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

      2. +-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-/r/ 99.0%

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

      6. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.1%

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

       2. *-rgt-identity 99.1%

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

       3. +-commutative 99.1%

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

       4. +-commutative 99.1%

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

       5. *-commutative 99.1%

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

       6. +-commutative 99.1%

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

       7. +-commutative 99.1%

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

       8. +-commutative 99.1%

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

       9. +-commutative 99.1%

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

       10. +-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in alpha around 0 85.6%

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

       1. +-commutative 85.6%

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

       2. *-commutative 85.6%

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

       3. +-commutative 85.6%

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

   14. Simplified 85.6%

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

   15. Taylor expanded in beta around inf 83.9%

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

       1. +-commutative 83.9%

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

   17. Simplified 83.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 160000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + 4}\\ \end{array} \]

Alternative 8: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.4e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (- alpha -1.0) (+ alpha (+ beta 2.0))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.4e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / (alpha + (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 <= 8.4d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (beta + 2.0d0))) / (alpha + (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 <= 8.4e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.4e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.4e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + 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 <= 8.4e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha - -1.0) / (alpha + (beta + 2.0))) / (alpha + (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, 8.4e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.4e15

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. associate-*r/ 99.4%

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

      2. +-commutative 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in alpha around 0 83.2%

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

      1. +-commutative 83.2%

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

   8. Simplified 83.2%

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

   9. Taylor expanded in alpha around 0 67.0%

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

   if 8.4e15 < beta

   1. Initial program 85.1%

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

      1. associate-/l/ 82.6%

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

      2. +-commutative 82.6%

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

      3. +-commutative 82.6%

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

      4. associate-+r+ 82.6%

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

      5. *-commutative 82.6%

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

      6. metadata-eval 82.6%

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

      7. associate-+l+ 82.6%

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

      8. metadata-eval 82.6%

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

      9. associate-+l+ 82.6%

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

      10. metadata-eval 82.6%

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

      11. metadata-eval 82.6%

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

      12. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in beta around -inf 86.6%

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

      1. expm1-log1p-u 86.6%

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

      2. expm1-udef 51.0%

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

      3. mul-1-neg 51.0%

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

      4. *-commutative 51.0%

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

      5. fma-neg 51.0%

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

      6. metadata-eval 51.0%

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

      7. *-commutative 51.0%

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

      8. associate-+r+ 51.0%

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

      9. +-commutative 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. expm1-def 86.6%

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

      2. expm1-log1p 86.6%

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

      3. associate-/r* 84.1%

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

   8. Simplified 84.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9: 97.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6)
   (/ (+ 0.5 (* beta 0.25)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = (0.5d0 + (beta * 0.25d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(Float64(0.5 + Float64(beta * 0.25)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5999999999999996

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in alpha around 0 82.9%

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

      1. +-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in beta around 0 82.3%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 82.3%

          \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   11. Simplified 82.3%

       \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   12. Taylor expanded in alpha around 0 65.8%

       \[\leadsto \frac{0.5 + \beta \cdot 0.25}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 4.5999999999999996 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in beta around inf 82.7%

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

      1. un-div-inv 82.8%

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

      2. +-commutative 82.8%

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

      3. +-commutative 82.8%

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

   6. Applied egg-rr 82.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \]

Alternative 10: 97.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.25:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.25)
   (/ (+ 0.5 (* beta 0.25)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ beta 4.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.25) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0);
	}
	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 <= 1.25d0) then
        tmp = (0.5d0 + (beta * 0.25d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (beta + 4.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.25) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.25:
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.25)
		tmp = Float64(Float64(0.5 + Float64(beta * 0.25)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(beta + 4.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.25)
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 4.0);
	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, 1.25], N[(N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in alpha around 0 82.9%

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

      1. +-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in beta around 0 82.3%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 82.3%

          \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   11. Simplified 82.3%

       \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   12. Taylor expanded in alpha around 0 65.8%

       \[\leadsto \frac{0.5 + \beta \cdot 0.25}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 1.25 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

      1. *-commutative 92.5%

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

      2. clear-num 92.4%

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

      3. clear-num 92.4%

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

      4. frac-times 92.5%

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

      5. metadata-eval 92.5%

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

      6. +-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 92.5%

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

      3. +-commutative 92.5%

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

      4. associate-/l* 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. div-inv 99.0%

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

      2. +-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-/r/ 99.0%

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

      6. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.1%

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

       2. *-rgt-identity 99.1%

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

       3. +-commutative 99.1%

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

       4. +-commutative 99.1%

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

       5. *-commutative 99.1%

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

       6. +-commutative 99.1%

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

       7. +-commutative 99.1%

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

       8. +-commutative 99.1%

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

       9. +-commutative 99.1%

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

       10. +-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in alpha around 0 85.7%

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

       1. +-commutative 85.7%

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

       2. *-commutative 85.7%

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

       3. +-commutative 85.7%

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

   14. Simplified 85.7%

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

   15. Taylor expanded in beta around inf 83.5%

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

       1. +-commutative 83.5%

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

   17. Simplified 83.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.25:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + 4}\\ \end{array} \]

Alternative 11: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.7)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.7d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.7:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.7)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.7)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.7], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in alpha around 0 82.9%

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

      1. +-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in beta around 0 81.8%

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

   if 3.7000000000000002 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in beta around inf 82.7%

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

      1. un-div-inv 82.8%

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

      2. +-commutative 82.8%

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

      3. +-commutative 82.8%

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

   6. Applied egg-rr 82.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \]

Alternative 12: 62.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{6 + \beta \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/ 1.0 (+ 6.0 (* beta 5.0)))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (6.0 + (beta * 5.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.4d0) then
        tmp = 1.0d0 / (6.0d0 + (beta * 5.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (6.0 + (beta * 5.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 1.0 / (6.0 + (beta * 5.0))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(1.0 / Float64(6.0 + Float64(beta * 5.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 1.0 / (6.0 + (beta * 5.0));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.4], N[(1.0 / N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. *-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+l+ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around -inf 28.6%

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

   5. Taylor expanded in alpha around 0 14.1%

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

      1. +-commutative 14.1%

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

      2. +-commutative 14.1%

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

   7. Simplified 14.1%

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

   8. Taylor expanded in beta around 0 14.1%

      \[\leadsto \frac{1}{\color{blue}{6 + 5 \cdot \beta}} \]
   9. Step-by-step derivation

      1. *-commutative 14.1%

         \[\leadsto \frac{1}{6 + \color{blue}{\beta \cdot 5}} \]

   10. Simplified 14.1%

       \[\leadsto \frac{1}{\color{blue}{6 + \beta \cdot 5}} \]

   if 6.4000000000000004 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in beta around inf 82.7%

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

   5. Taylor expanded in beta around inf 82.4%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{6 + \beta \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 13: 96.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in alpha around 0 82.9%

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

      1. +-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in beta around 0 81.8%

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

   if 4.5 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in beta around inf 82.7%

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

   5. Taylor expanded in beta around inf 82.4%

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

4. Final simplification 82.0%

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

Alternative 14: 57.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.1)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ 1.0 (* beta (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.1d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.1:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = 1.0 / (beta * (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.1)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.1)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = 1.0 / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.1], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1000000000000001

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. *-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+l+ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around -inf 28.6%

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

   5. Taylor expanded in alpha around 0 14.1%

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

      1. +-commutative 14.1%

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

      2. +-commutative 14.1%

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

   7. Simplified 14.1%

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

   8. Taylor expanded in beta around 0 14.1%

      \[\leadsto \color{blue}{0.16666666666666666 + -0.1388888888888889 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 14.1%

         \[\leadsto 0.16666666666666666 + \color{blue}{\beta \cdot -0.1388888888888889} \]

   10. Simplified 14.1%

       \[\leadsto \color{blue}{0.16666666666666666 + \beta \cdot -0.1388888888888889} \]

   if 1.1000000000000001 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in beta around inf 82.7%

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

   5. Taylor expanded in alpha around 0 79.0%

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

      1. +-commutative 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 15: 57.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1}{6 + \beta \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5) (/ 1.0 (+ 6.0 (* beta 5.0))) (/ 1.0 (* beta (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 1.0 / (6.0 + (beta * 5.0));
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 1.0d0 / (6.0d0 + (beta * 5.0d0))
    else
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 1.0 / (6.0 + (beta * 5.0));
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 1.0 / (6.0 + (beta * 5.0))
	else:
		tmp = 1.0 / (beta * (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(1.0 / Float64(6.0 + Float64(beta * 5.0)));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 1.0 / (6.0 + (beta * 5.0));
	else
		tmp = 1.0 / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(1.0 / N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. *-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+l+ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around -inf 28.6%

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

   5. Taylor expanded in alpha around 0 14.1%

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

      1. +-commutative 14.1%

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

      2. +-commutative 14.1%

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

   7. Simplified 14.1%

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

   8. Taylor expanded in beta around 0 14.1%

      \[\leadsto \frac{1}{\color{blue}{6 + 5 \cdot \beta}} \]
   9. Step-by-step derivation

      1. *-commutative 14.1%

         \[\leadsto \frac{1}{6 + \color{blue}{\beta \cdot 5}} \]

   10. Simplified 14.1%

       \[\leadsto \frac{1}{\color{blue}{6 + \beta \cdot 5}} \]

   if 4.5 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in beta around inf 82.7%

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

   5. Taylor expanded in alpha around 0 79.0%

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

      1. +-commutative 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1}{6 + \beta \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 16: 12.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35:\\ \;\;\;\;0.16666666666666666 + \alpha \cdot 0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.35)
   (+ 0.16666666666666666 (* alpha 0.027777777777777776))
   (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35) {
		tmp = 0.16666666666666666 + (alpha * 0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.35d0) then
        tmp = 0.16666666666666666d0 + (alpha * 0.027777777777777776d0)
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35) {
		tmp = 0.16666666666666666 + (alpha * 0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.35:
		tmp = 0.16666666666666666 + (alpha * 0.027777777777777776)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.35)
		tmp = Float64(0.16666666666666666 + Float64(alpha * 0.027777777777777776));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.35)
		tmp = 0.16666666666666666 + (alpha * 0.027777777777777776);
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.35], N[(0.16666666666666666 + N[(alpha * 0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3500000000000001

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. *-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+l+ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around -inf 28.6%

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

   5. Taylor expanded in beta around 0 28.5%

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

      1. associate-*r/ 28.5%

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

      2. neg-mul-1 28.5%

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

      3. sub-neg 28.5%

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

      4. mul-1-neg 28.5%

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

      5. distribute-neg-in 28.5%

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

      6. +-commutative 28.5%

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

      7. mul-1-neg 28.5%

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

      8. distribute-lft-in 28.5%

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

      9. metadata-eval 28.5%

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

      10. mul-1-neg 28.5%

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

      11. unsub-neg 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in alpha around 0 13.7%

      \[\leadsto \color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \alpha} \]
   9. Step-by-step derivation

      1. *-commutative 13.7%

         \[\leadsto 0.16666666666666666 + \color{blue}{\alpha \cdot 0.027777777777777776} \]

   10. Simplified 13.7%

       \[\leadsto \color{blue}{0.16666666666666666 + \alpha \cdot 0.027777777777777776} \]

   if 1.3500000000000001 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

      1. associate-*r/ 92.6%

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

      2. +-commutative 92.6%

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

   5. Applied egg-rr 92.6%

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

   6. Taylor expanded in alpha around 0 81.4%

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

      1. +-commutative 81.4%

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

   8. Simplified 81.4%

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

   9. Taylor expanded in beta around 0 50.8%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 50.8%

          \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   11. Simplified 50.8%

       \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   12. Taylor expanded in beta around inf 6.9%

       \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35:\\ \;\;\;\;0.16666666666666666 + \alpha \cdot 0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 17: 12.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.1)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.1d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.1:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.1)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.1)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.1], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1000000000000001

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. *-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+l+ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around -inf 28.6%

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

   5. Taylor expanded in alpha around 0 14.1%

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

      1. +-commutative 14.1%

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

      2. +-commutative 14.1%

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

   7. Simplified 14.1%

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

   8. Taylor expanded in beta around 0 14.1%

      \[\leadsto \color{blue}{0.16666666666666666 + -0.1388888888888889 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 14.1%

         \[\leadsto 0.16666666666666666 + \color{blue}{\beta \cdot -0.1388888888888889} \]

   10. Simplified 14.1%

       \[\leadsto \color{blue}{0.16666666666666666 + \beta \cdot -0.1388888888888889} \]

   if 1.1000000000000001 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

      1. associate-*r/ 92.6%

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

      2. +-commutative 92.6%

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

   5. Applied egg-rr 92.6%

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

   6. Taylor expanded in alpha around 0 81.4%

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

      1. +-commutative 81.4%

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

   8. Simplified 81.4%

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

   9. Taylor expanded in beta around 0 50.8%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 50.8%

          \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   11. Simplified 50.8%

       \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   12. Taylor expanded in beta around inf 6.9%

       \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 18: 12.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5) 0.16666666666666666 (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d0) then
        tmp = 0.16666666666666666d0
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = 0.16666666666666666
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = 0.16666666666666666;
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = 0.16666666666666666;
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5], 0.16666666666666666, N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   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/ 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. *-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+l+ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around -inf 28.6%

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

   5. Taylor expanded in beta around 0 28.5%

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

      1. associate-*r/ 28.5%

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

      2. neg-mul-1 28.5%

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

      3. sub-neg 28.5%

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

      4. mul-1-neg 28.5%

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

      5. distribute-neg-in 28.5%

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

      6. +-commutative 28.5%

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

      7. mul-1-neg 28.5%

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

      8. distribute-lft-in 28.5%

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

      9. metadata-eval 28.5%

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

      10. mul-1-neg 28.5%

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

      11. unsub-neg 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in alpha around 0 14.1%

      \[\leadsto \color{blue}{0.16666666666666666} \]

   if 1.5 < beta

   1. Initial program 85.6%

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

   3. Simplified 92.5%

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

      1. associate-*r/ 92.6%

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

      2. +-commutative 92.6%

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

   5. Applied egg-rr 92.6%

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

   6. Taylor expanded in alpha around 0 81.4%

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

      1. +-commutative 81.4%

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

   8. Simplified 81.4%

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

   9. Taylor expanded in beta around 0 50.8%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 50.8%

          \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   11. Simplified 50.8%

       \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   12. Taylor expanded in beta around inf 6.9%

       \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 19: 10.5% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.16666666666666666 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.16666666666666666)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.16666666666666666
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.16666666666666666

Derivation

1. Initial program 95.2%

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

   1. associate-/l/ 94.2%

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

   2. +-commutative 94.2%

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

   3. +-commutative 94.2%

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

   4. associate-+r+ 94.2%

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

   5. *-commutative 94.2%

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

   6. metadata-eval 94.2%

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

   7. associate-+l+ 94.2%

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

   8. metadata-eval 94.2%

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

   9. associate-+l+ 94.2%

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

   10. metadata-eval 94.2%

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

   11. metadata-eval 94.2%

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

   12. associate-+l+ 94.2%

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

3. Simplified 94.2%

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

4. Taylor expanded in beta around -inf 47.1%

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

5. Taylor expanded in beta around 0 23.6%

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

   1. associate-*r/ 23.6%

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

   2. neg-mul-1 23.6%

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

   3. sub-neg 23.6%

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

   4. mul-1-neg 23.6%

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

   5. distribute-neg-in 23.6%

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

   6. +-commutative 23.6%

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

   7. mul-1-neg 23.6%

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

   8. distribute-lft-in 23.6%

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

   9. metadata-eval 23.6%

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

   10. mul-1-neg 23.6%

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

   11. unsub-neg 23.6%

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

7. Simplified 23.6%

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

8. Taylor expanded in alpha around 0 10.8%

   \[\leadsto \color{blue}{0.16666666666666666} \]

9. Final simplification 10.8%

   \[\leadsto 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023331 
(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)))