Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 21.8s

Alternatives: 20

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 97.3%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

8. Simplified 99.7%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. *-commutative 99.8%

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

    2. clear-num 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-+r+ 99.8%

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

    5. frac-times 99.7%

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

    6. *-un-lft-identity 99.7%

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

    7. +-commutative 99.7%

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

    8. +-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.28 \cdot 10^{+14}:\\ \;\;\;\;\frac{\alpha + 1}{t_0} \cdot \frac{\beta + 1}{t_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 4\right) + \alpha \cdot 2}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.28e+14)
     (* (/ (+ alpha 1.0) t_0) (/ (+ beta 1.0) (* t_0 (+ alpha (+ beta 3.0)))))
     (/
      (/ (+ alpha 1.0) (+ (+ beta 4.0) (* alpha 2.0)))
      (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.28e+14) {
		tmp = ((alpha + 1.0) / t_0) * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.28d+14) then
        tmp = ((alpha + 1.0d0) / t_0) * ((beta + 1.0d0) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) / ((beta + 4.0d0) + (alpha * 2.0d0))) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.28e+14) {
		tmp = ((alpha + 1.0) / t_0) * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.28e+14:
		tmp = ((alpha + 1.0) / t_0) * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / (2.0 + (alpha + 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 <= 1.28e+14)
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(beta + 1.0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0))) / Float64(2.0 + Float64(alpha + 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 <= 1.28e+14)
		tmp = ((alpha + 1.0) / t_0) * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.28e+14], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.28e14

   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 99.8%

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

   if 1.28e14 < beta

   1. Initial program 84.2%

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

   3. Simplified 92.1%

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

      1. associate-*r/ 92.2%

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

      2. +-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. clear-num 99.8%

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

       3. +-commutative 99.8%

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

       4. associate-+r+ 99.8%

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

       5. frac-times 99.5%

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

       6. *-un-lft-identity 99.5%

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

       7. +-commutative 99.5%

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

       8. +-commutative 99.5%

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

   12. Applied egg-rr 99.5%

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

   13. Taylor expanded in beta around inf 80.1%

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

       1. associate-+r+ 80.1%

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

   15. Simplified 80.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.28 \cdot 10^{+14}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 4\right) + \alpha \cdot 2}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 50000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 3} \cdot \frac{1}{\beta + 2}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{t_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \end{array} \]

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 50000000.0)
     (/
      (* (/ (+ beta 1.0) (+ beta 3.0)) (/ 1.0 (+ beta 2.0)))
      (+ 2.0 (+ alpha beta)))
     (* (/ (/ (+ alpha 1.0) t_0) t_0) (- 1.0 (/ (+ alpha 2.0) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 50000000.0) {
		tmp = (((beta + 1.0) / (beta + 3.0)) * (1.0 / (beta + 2.0))) / (2.0 + (alpha + beta));
	} else {
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 50000000.0d0) then
        tmp = (((beta + 1.0d0) / (beta + 3.0d0)) * (1.0d0 / (beta + 2.0d0))) / (2.0d0 + (alpha + beta))
    else
        tmp = (((alpha + 1.0d0) / t_0) / t_0) * (1.0d0 - ((alpha + 2.0d0) / 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 <= 50000000.0) {
		tmp = (((beta + 1.0) / (beta + 3.0)) * (1.0 / (beta + 2.0))) / (2.0 + (alpha + beta));
	} else {
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 50000000.0:
		tmp = (((beta + 1.0) / (beta + 3.0)) * (1.0 / (beta + 2.0))) / (2.0 + (alpha + beta))
	else:
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / 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 <= 50000000.0)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(beta + 3.0)) * Float64(1.0 / Float64(beta + 2.0))) / Float64(2.0 + Float64(alpha + beta)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / t_0) / t_0) * Float64(1.0 - Float64(Float64(alpha + 2.0) / 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 <= 50000000.0)
		tmp = (((beta + 1.0) / (beta + 3.0)) * (1.0 / (beta + 2.0))) / (2.0 + (alpha + beta));
	else
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 50000000.0], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e7

   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 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in alpha around 0 65.3%

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

       1. +-commutative 65.3%

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

   13. Simplified 65.3%

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

   14. Taylor expanded in alpha around 0 65.4%

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

       1. +-commutative 65.4%

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

   16. Simplified 65.4%

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

   if 5e7 < beta

   1. Initial program 84.6%

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

   3. Simplified 92.3%

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

      1. associate-*r/ 92.4%

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

      2. +-commutative 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around inf 78.5%

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

       1. mul-1-neg 78.5%

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

       2. unsub-neg 78.5%

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

   11. Simplified 78.5%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 50000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 3} \cdot \frac{1}{\beta + 2}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 97.3%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 5: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 97.3%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

8. Simplified 99.7%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 6: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 97.3%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in alpha around 0 69.7%

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

    1. +-commutative 69.7%

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

11. Simplified 69.7%

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

12. Final simplification 69.7%

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

Alternative 7: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 97.3%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

8. Simplified 99.7%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Taylor expanded in alpha around 0 69.7%

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

    1. +-commutative 69.7%

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

13. Simplified 69.7%

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

14. Final simplification 69.7%

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

Alternative 8: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 6.8)
     (/ (/ (+ alpha 1.0) (* (+ alpha 3.0) (+ alpha 2.0))) t_0)
     (/ (/ (+ alpha 1.0) beta) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 6.8) {
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / (t_0 + 1.0);
	}
	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 = 2.0d0 + (alpha + beta)
    if (beta <= 6.8d0) then
        tmp = ((alpha + 1.0d0) / ((alpha + 3.0d0) * (alpha + 2.0d0))) / t_0
    else
        tmp = ((alpha + 1.0d0) / beta) / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 6.8) {
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / (t_0 + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 6.8:
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0
	else:
		tmp = ((alpha + 1.0) / beta) / (t_0 + 1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 6.8)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 6.8)
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0;
	else
		tmp = ((alpha + 1.0) / beta) / (t_0 + 1.0);
	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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.8], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.79999999999999982

   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 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around 0 97.3%

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

   if 6.79999999999999982 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around -inf 77.2%

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

      1. associate-*r/ 77.2%

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

      2. mul-1-neg 77.2%

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

      3. sub-neg 77.2%

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

      4. mul-1-neg 77.2%

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

      5. distribute-neg-in 77.2%

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

      6. +-commutative 77.2%

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

      7. mul-1-neg 77.2%

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

      8. distribute-lft-in 77.2%

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

      9. metadata-eval 77.2%

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

      10. mul-1-neg 77.2%

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

      11. unsub-neg 77.2%

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

   5. Simplified 77.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 4\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.5)
     (/ (/ (+ alpha 1.0) (* (+ alpha 3.0) (+ alpha 2.0))) t_0)
     (/ (/ (+ alpha 1.0) (+ (+ beta 4.0) (* alpha 2.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.5) {
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 3.5d0) then
        tmp = ((alpha + 1.0d0) / ((alpha + 3.0d0) * (alpha + 2.0d0))) / t_0
    else
        tmp = ((alpha + 1.0d0) / ((beta + 4.0d0) + (alpha * 2.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.5) {
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 3.5:
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0
	else:
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.5)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 3.5)
		tmp = ((alpha + 1.0) / ((alpha + 3.0) * (alpha + 2.0))) / t_0;
	else
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.5], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5

   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 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around 0 97.3%

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

   if 3.5 < beta

   1. Initial program 84.7%

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

   3. Simplified 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. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 92.5%

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

      2. +-commutative 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. clear-num 99.8%

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

       3. +-commutative 99.8%

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

       4. associate-+r+ 99.8%

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

       5. frac-times 99.5%

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

       6. *-un-lft-identity 99.5%

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

       7. +-commutative 99.5%

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

       8. +-commutative 99.5%

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

   12. Applied egg-rr 99.5%

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

   13. Taylor expanded in beta around inf 78.9%

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

       1. associate-+r+ 78.9%

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

   15. Simplified 78.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 4\right) + \alpha \cdot 2}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 17200000000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 4\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 17200000000000.0)
     (/ (/ (+ beta 1.0) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/ (/ (+ alpha 1.0) (+ (+ beta 4.0) (* alpha 2.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 17200000000000.0) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 17200000000000.0d0) then
        tmp = ((beta + 1.0d0) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = ((alpha + 1.0d0) / ((beta + 4.0d0) + (alpha * 2.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 17200000000000.0) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 17200000000000.0:
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 17200000000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 17200000000000.0)
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = ((alpha + 1.0) / ((beta + 4.0) + (alpha * 2.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 17200000000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.72e13

   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 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in alpha around 0 65.3%

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

       1. +-commutative 65.3%

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

       2. +-commutative 65.3%

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

   13. Simplified 65.3%

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

   if 1.72e13 < beta

   1. Initial program 84.2%

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

   3. Simplified 92.1%

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

      1. associate-*r/ 92.2%

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

      2. +-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. clear-num 99.8%

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

       3. +-commutative 99.8%

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

       4. associate-+r+ 99.8%

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

       5. frac-times 99.5%

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

       6. *-un-lft-identity 99.5%

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

       7. +-commutative 99.5%

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

       8. +-commutative 99.5%

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

   12. Applied egg-rr 99.5%

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

   13. Taylor expanded in beta around inf 80.1%

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

       1. associate-+r+ 80.1%

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

   15. Simplified 80.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 17200000000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 4\right) + \alpha \cdot 2}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.8)
   (*
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (+ alpha 1.0) (+ 2.0 (+ alpha beta))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.8d0) then
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((alpha + 1.0d0) / (2.0d0 + (alpha + beta))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.8:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.8)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(alpha + beta))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.8)
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.8], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999982

   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 99.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in alpha around 0 64.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   9. Step-by-step derivation

      1. *-commutative 64.4%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   10. Simplified 64.4%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   if 5.79999999999999982 < beta

   1. Initial program 84.7%

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

   3. Simplified 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. Add Preprocessing

   5. Taylor expanded in beta around inf 77.0%

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

      1. un-div-inv 77.2%

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

      2. +-commutative 77.2%

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

      3. associate-+r+ 77.2%

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

      4. +-commutative 77.2%

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

   7. Applied egg-rr 77.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.1:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.1)
   (*
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (+ alpha 1.0) beta) (+ (+ 2.0 (+ alpha beta)) 1.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.1) {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((2.0 + (alpha + beta)) + 1.0);
	}
	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 <= 5.1d0) then
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / ((2.0d0 + (alpha + beta)) + 1.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.1) {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((2.0 + (alpha + beta)) + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.1:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((alpha + 1.0) / beta) / ((2.0 + (alpha + beta)) + 1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.1)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(2.0 + Float64(alpha + beta)) + 1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.1)
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((alpha + 1.0) / beta) / ((2.0 + (alpha + beta)) + 1.0);
	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, 5.1], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0999999999999996

   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 99.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in alpha around 0 64.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   9. Step-by-step derivation

      1. *-commutative 64.4%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   10. Simplified 64.4%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   if 5.0999999999999996 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around -inf 77.2%

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

      1. associate-*r/ 77.2%

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

      2. mul-1-neg 77.2%

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

      3. sub-neg 77.2%

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

      4. mul-1-neg 77.2%

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

      5. distribute-neg-in 77.2%

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

      6. +-commutative 77.2%

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

      7. mul-1-neg 77.2%

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

      8. distribute-lft-in 77.2%

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

      9. metadata-eval 77.2%

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

      10. mul-1-neg 77.2%

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

      11. unsub-neg 77.2%

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

   5. Simplified 77.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.1:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 7.4 \cdot 10^{+159}:\\ \;\;\;\;\frac{-1}{\beta} \cdot \frac{-1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 0.16666666666666666 (+ beta 2.0))
   (if (<= beta 7.4e+159)
     (* (/ -1.0 beta) (/ -1.0 (+ beta 2.0)))
     (* (/ 1.0 beta) (/ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else if (beta <= 7.4e+159) {
		tmp = (-1.0 / beta) * (-1.0 / (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else if (beta <= 7.4d+159) then
        tmp = ((-1.0d0) / beta) * ((-1.0d0) / (beta + 2.0d0))
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else if (beta <= 7.4e+159) {
		tmp = (-1.0 / beta) * (-1.0 / (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.16666666666666666 / (beta + 2.0)
	elif beta <= 7.4e+159:
		tmp = (-1.0 / beta) * (-1.0 / (beta + 2.0))
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	elseif (beta <= 7.4e+159)
		tmp = Float64(Float64(-1.0 / beta) * Float64(-1.0 / Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666 / (beta + 2.0);
	elseif (beta <= 7.4e+159)
		tmp = (-1.0 / beta) * (-1.0 / (beta + 2.0));
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 7.4e+159], N[(N[(-1.0 / beta), $MachinePrecision] * N[(-1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6

   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 99.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 6 < beta < 7.40000000000000002e159

   1. Initial program 89.6%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in beta around inf 67.4%

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

   6. Taylor expanded in alpha around 0 62.0%

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

   if 7.40000000000000002e159 < beta

   1. Initial program 78.1%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in beta around inf 87.4%

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

   6. Taylor expanded in alpha around inf 87.4%

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

   7. Taylor expanded in beta around inf 90.0%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 7.4 \cdot 10^{+159}:\\ \;\;\;\;\frac{-1}{\beta} \cdot \frac{-1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 93.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 0.16666666666666666 (+ beta 2.0))
   (if (<= beta 1.35e+154)
     (/ 1.0 (* beta (+ beta 2.0)))
     (* (/ 1.0 beta) (/ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else if (beta <= 1.35e+154) {
		tmp = 1.0 / (beta * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else if (beta <= 1.35d+154) then
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else if (beta <= 1.35e+154) {
		tmp = 1.0 / (beta * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.16666666666666666 / (beta + 2.0)
	elif beta <= 1.35e+154:
		tmp = 1.0 / (beta * (beta + 2.0))
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	elseif (beta <= 1.35e+154)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666 / (beta + 2.0);
	elseif (beta <= 1.35e+154)
		tmp = 1.0 / (beta * (beta + 2.0));
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6

   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 99.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 6 < beta < 1.35000000000000003e154

   1. Initial program 89.4%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in beta around inf 66.7%

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

   6. Taylor expanded in alpha around 0 61.3%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 78.7%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in beta around inf 86.4%

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

   6. Taylor expanded in alpha around inf 86.4%

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

   7. Taylor expanded in beta around inf 88.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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 99.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 6 < beta

   1. Initial program 84.7%

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

   3. Simplified 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. Add Preprocessing

   5. Taylor expanded in beta around inf 77.0%

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

      1. un-div-inv 77.2%

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

      2. +-commutative 77.2%

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

      3. associate-+r+ 77.2%

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

      4. +-commutative 77.2%

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

   7. Applied egg-rr 77.2%

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

4. Final simplification 67.9%

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

Alternative 16: 96.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((alpha + 1.0) / 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 <= 8.0d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((alpha + 1.0d0) / 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 <= 8.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if beta < 8

   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 99.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 8 < beta

   1. Initial program 84.7%

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

   3. Simplified 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. Add Preprocessing

   5. Taylor expanded in beta around inf 77.0%

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

   6. Taylor expanded in beta around inf 76.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8e+56)
   (/ 0.16666666666666666 (+ beta 2.0))
   (* (/ 1.0 beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8e+56) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.8d+56) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8e+56) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.8e+56:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8e+56)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.8e+56)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = (1.0 / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.8e+56], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.79999999999999999e56

   1. Initial program 99.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 91.3%

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

      1. +-commutative 91.3%

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

   7. Simplified 91.3%

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

   8. Taylor expanded in alpha around 0 57.9%

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

      1. +-commutative 57.9%

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

   10. Simplified 57.9%

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

   if 1.79999999999999999e56 < beta

   1. Initial program 82.2%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in beta around inf 87.1%

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

   6. Taylor expanded in alpha around inf 58.2%

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

   7. Taylor expanded in beta around inf 55.4%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 195000000000:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 195000000000.0)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ 1.0 (* alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 195000000000.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (alpha * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 195000000000.0d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = 1.0d0 / (alpha * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 195000000000.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (alpha * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 195000000000.0:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = 1.0 / (alpha * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 195000000000.0)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(1.0 / Float64(alpha * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 195000000000.0)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = 1.0 / (alpha * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 195000000000.0], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(alpha * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.95e11

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in beta around 0 67.8%

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

      1. +-commutative 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in alpha around 0 63.9%

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

      1. +-commutative 63.9%

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

   10. Simplified 63.9%

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

   if 1.95e11 < alpha

   1. Initial program 84.6%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in beta around inf 26.3%

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

   6. Taylor expanded in beta around 0 51.0%

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

   7. Taylor expanded in alpha around inf 13.6%

      \[\leadsto \color{blue}{\frac{1}{\alpha \cdot \beta}} \]
   8. Step-by-step derivation

      1. *-commutative 13.6%

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

   9. Simplified 13.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 195000000000:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 47.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\beta + 2} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Fortran

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

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (beta + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(beta + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (beta + 2.0);
end

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 97.3%

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

5. Taylor expanded in beta around 0 71.7%

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

   1. +-commutative 71.7%

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

7. Simplified 71.7%

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

8. Taylor expanded in alpha around 0 44.5%

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

   1. +-commutative 44.5%

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

10. Simplified 44.5%

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

11. Final simplification 44.5%

    \[\leadsto \frac{0.16666666666666666}{\beta + 2} \]
12. Add Preprocessing

Alternative 20: 6.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / beta)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 97.3%

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

5. Taylor expanded in beta around inf 31.5%

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

6. Taylor expanded in beta around 0 19.5%

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

7. Taylor expanded in alpha around 0 4.0%

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

8. Final simplification 4.0%

   \[\leadsto \frac{0.16666666666666666}{\beta} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))