Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.8%

Time: 14.3s

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: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 83.8%

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

   1. times-frac 96.7%

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

   2. +-commutative 96.7%

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

6. Applied egg-rr 96.7%

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

   1. +-commutative 96.7%

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

   2. +-commutative 96.7%

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

   3. +-commutative 96.7%

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

   4. +-commutative 96.7%

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

   5. +-commutative 96.7%

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

   6. +-commutative 96.7%

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

   7. +-commutative 96.7%

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

   8. associate-+r+ 96.7%

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

   9. +-commutative 96.7%

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

   10. +-commutative 96.7%

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

8. Simplified 96.7%

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

   1. associate-*r/ 96.8%

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

   2. +-commutative 96.8%

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

   3. +-commutative 96.8%

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

   4. *-commutative 96.8%

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

   5. +-commutative 96.8%

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

   6. +-commutative 96.8%

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

   7. associate-+r+ 96.7%

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

10. Applied egg-rr 96.7%

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

    1. associate-/r* 99.8%

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

    2. *-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

    5. +-commutative 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 + \beta}{t\_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \frac{\alpha}{t\_0}}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.35e+154)
     (* (/ (+ 1.0 alpha) t_0) (/ (+ 1.0 beta) (* t_0 (+ 3.0 (+ beta alpha)))))
     (/ (/ (* (+ 1.0 beta) (/ alpha t_0)) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.35e+154) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (3.0 + (beta + alpha))));
	} else {
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / 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 = alpha + (beta + 2.0d0)
    if (beta <= 1.35d+154) then
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 + beta) / (t_0 * (3.0d0 + (beta + alpha))))
    else
        tmp = (((1.0d0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0d0))) / t_0
    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 <= 1.35e+154) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (3.0 + (beta + alpha))));
	} else {
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.35e+154:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (3.0 + (beta + alpha))))
	else:
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.35e+154)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(3.0 + Float64(beta + alpha)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(alpha / t_0)) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.35e+154)
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (3.0 + (beta + alpha))));
	else
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.35e+154], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.35000000000000003e154

   1. Initial program 98.4%

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

   3. Simplified 85.9%

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

      1. times-frac 98.9%

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

      2. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. +-commutative 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. +-commutative 98.9%

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

      5. +-commutative 98.9%

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

      6. +-commutative 98.9%

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

      7. +-commutative 98.9%

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

      8. associate-+r+ 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

   8. Simplified 98.9%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 75.4%

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

   3. Simplified 71.1%

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

      1. times-frac 83.5%

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

      2. +-commutative 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. +-commutative 83.5%

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

      2. +-commutative 83.5%

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

      3. +-commutative 83.5%

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

      4. +-commutative 83.5%

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

      5. +-commutative 83.5%

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

      6. +-commutative 83.5%

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

      7. +-commutative 83.5%

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

      8. associate-+r+ 83.5%

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

      9. +-commutative 83.5%

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

      10. +-commutative 83.5%

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

   8. Simplified 83.5%

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

      1. associate-*r/ 83.5%

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

      2. +-commutative 83.5%

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

      3. +-commutative 83.5%

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

      4. *-commutative 83.5%

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

      5. +-commutative 83.5%

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

      6. +-commutative 83.5%

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

      7. associate-+r+ 83.5%

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

   10. Applied egg-rr 83.5%

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

       1. associate-/r* 99.8%

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

       2. *-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around inf 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \frac{\alpha}{t\_0}}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= alpha 2.3e+54)
     (* (/ (+ 1.0 alpha) t_0) (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (* (+ 1.0 beta) (/ alpha t_0)) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (alpha <= 2.3e+54) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / 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 = alpha + (beta + 2.0d0)
    if (alpha <= 2.3d+54) then
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = (((1.0d0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0d0))) / t_0
    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 (alpha <= 2.3e+54) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if alpha <= 2.3e+54:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (alpha <= 2.3e+54)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(alpha / t_0)) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (alpha <= 2.3e+54)
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = (((1.0 + beta) * (alpha / t_0)) / (alpha + (beta + 3.0))) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.3e+54], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.29999999999999994e54

   1. Initial program 99.2%

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

   3. Simplified 89.8%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+r+ 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 93.7%

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

       1. +-commutative 93.7%

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

       2. *-commutative 93.7%

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

       3. +-commutative 93.7%

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

   11. Simplified 93.7%

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

   if 2.29999999999999994e54 < alpha

   1. Initial program 83.9%

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

   3. Simplified 67.3%

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

      1. times-frac 89.0%

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

      2. +-commutative 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

      2. +-commutative 89.0%

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

      3. +-commutative 89.0%

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

      4. +-commutative 89.0%

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

      5. +-commutative 89.0%

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

      6. +-commutative 89.0%

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

      7. +-commutative 89.0%

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

      8. associate-+r+ 89.0%

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

      9. +-commutative 89.0%

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

      10. +-commutative 89.0%

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

   8. Simplified 89.0%

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

      1. associate-*r/ 89.0%

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

      2. +-commutative 89.0%

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

      3. +-commutative 89.0%

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

      4. *-commutative 89.0%

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

      5. +-commutative 89.0%

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

      6. +-commutative 89.0%

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

      7. associate-+r+ 89.0%

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

   10. Applied egg-rr 89.0%

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

       1. associate-/r* 99.7%

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

       2. *-commutative 99.7%

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

       3. +-commutative 99.7%

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

       4. +-commutative 99.7%

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

       5. +-commutative 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in alpha around inf 99.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 230000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\beta \cdot \left(1 + \frac{2}{\beta}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 230000000.0)
   (/
    (/ (+ 1.0 beta) (+ beta 2.0))
    (* (+ 3.0 (+ beta alpha)) (* beta (+ 1.0 (/ 2.0 beta)))))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 230000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * (beta * (1.0 + (2.0 / beta))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 230000000.0d0) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((3.0d0 + (beta + alpha)) * (beta * (1.0d0 + (2.0d0 / beta))))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 230000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * (beta * (1.0 + (2.0 / beta))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 230000000.0:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * (beta * (1.0 + (2.0 / beta))))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 230000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(3.0 + Float64(beta + alpha)) * Float64(beta * Float64(1.0 + Float64(2.0 / beta)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 230000000.0)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * (beta * (1.0 + (2.0 / beta))));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 230000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(beta * N[(1.0 + N[(2.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3e8

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. metadata-eval 85.2%

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

   10. Simplified 85.2%

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

   11. Taylor expanded in alpha around 0 68.5%

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

       1. associate-*r/ 68.5%

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

       2. metadata-eval 68.5%

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

   13. Simplified 68.5%

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

   if 2.3e8 < beta

   1. Initial program 87.1%

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

   3. Simplified 62.1%

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

      1. times-frac 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. +-commutative 91.4%

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

      2. +-commutative 91.4%

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

      3. +-commutative 91.4%

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

      4. +-commutative 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

      7. +-commutative 91.4%

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

      8. associate-+r+ 91.4%

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

      9. +-commutative 91.4%

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

      10. +-commutative 91.4%

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

   8. Simplified 91.4%

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

   9. Taylor expanded in beta around inf 83.8%

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

       1. mul-1-neg 83.8%

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

       2. unsub-neg 83.8%

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

   11. Simplified 83.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 230000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\beta \cdot \left(1 + \frac{2}{\beta}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 67000000000:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 67000000000.0)
     (/ (* (+ 1.0 beta) (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0))) t_0)
     (*
      (/ (+ 1.0 alpha) t_0)
      (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 67000000000.0) {
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 67000000000.0d0) then
        tmp = ((1.0d0 + beta) * ((1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 67000000000.0) {
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 67000000000.0:
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / 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 <= 67000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 67000000000.0)
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 67000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.7e10

   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 96.4%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-/r* 99.8%

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

       2. *-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

   12. Simplified 99.8%

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

       1. div-inv 99.8%

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

       2. associate-+l+ 99.8%

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

   14. Applied egg-rr 99.8%

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

       1. associate-*l* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. associate-+r+ 99.8%

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

   16. Simplified 99.8%

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

   17. Taylor expanded in alpha around 0 68.1%

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

       1. associate-/r* 68.1%

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

   19. Simplified 68.1%

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

   if 6.7e10 < beta

   1. Initial program 87.1%

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

   3. Simplified 62.1%

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

      1. times-frac 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. +-commutative 91.4%

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

      2. +-commutative 91.4%

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

      3. +-commutative 91.4%

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

      4. +-commutative 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

      7. +-commutative 91.4%

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

      8. associate-+r+ 91.4%

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

      9. +-commutative 91.4%

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

      10. +-commutative 91.4%

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

   8. Simplified 91.4%

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

   9. Taylor expanded in beta around inf 83.8%

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

       1. mul-1-neg 83.8%

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

       2. unsub-neg 83.8%

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

   11. Simplified 83.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 67000000000:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 83.8%

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

   1. times-frac 96.7%

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

   2. +-commutative 96.7%

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

6. Applied egg-rr 96.7%

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

   1. +-commutative 96.7%

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

   2. +-commutative 96.7%

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

   3. +-commutative 96.7%

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

   4. +-commutative 96.7%

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

   5. +-commutative 96.7%

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

   6. +-commutative 96.7%

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

   7. +-commutative 96.7%

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

   8. associate-+r+ 96.7%

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

   9. +-commutative 96.7%

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

   10. +-commutative 96.7%

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

8. Simplified 96.7%

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

   1. associate-*l/ 96.8%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 7: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 8e+14)
     (/ (* (+ 1.0 beta) (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0))) t_0)
     (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 8e+14) {
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / 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 = alpha + (beta + 2.0d0)
    if (beta <= 8d+14) then
        tmp = ((1.0d0 + beta) * ((1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0
    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 <= 8e+14) {
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 8e+14:
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 8e+14)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 8e+14)
		tmp = ((1.0 + beta) * ((1.0 / (beta + 2.0)) / (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8e+14], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8e14

   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 96.4%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-/r* 99.8%

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

       2. *-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

   12. Simplified 99.8%

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

       1. div-inv 99.8%

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

       2. associate-+l+ 99.8%

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

   14. Applied egg-rr 99.8%

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

       1. associate-*l* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. associate-+r+ 99.9%

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

   16. Simplified 99.9%

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

   17. Taylor expanded in alpha around 0 68.3%

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

       1. associate-/r* 68.3%

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

   19. Simplified 68.3%

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

   if 8e14 < beta

   1. Initial program 87.0%

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

   3. Simplified 61.7%

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

      1. times-frac 91.3%

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

      2. +-commutative 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. +-commutative 91.3%

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

      2. +-commutative 91.3%

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

      3. +-commutative 91.3%

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

      4. +-commutative 91.3%

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

      5. +-commutative 91.3%

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

      6. +-commutative 91.3%

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

      7. +-commutative 91.3%

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

      8. associate-+r+ 91.3%

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

      9. +-commutative 91.3%

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

      10. +-commutative 91.3%

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

   8. Simplified 91.3%

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

      1. associate-*r/ 91.4%

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

      2. +-commutative 91.4%

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

      3. +-commutative 91.4%

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

      4. *-commutative 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

      7. associate-+r+ 91.4%

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

   10. Applied egg-rr 91.4%

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

       1. associate-/r* 99.6%

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

       2. *-commutative 99.6%

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

       3. +-commutative 99.6%

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

       4. +-commutative 99.6%

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

       5. +-commutative 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in beta around inf 84.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 4.15e+15)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.15e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / 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 = alpha + (beta + 2.0d0)
    if (beta <= 4.15d+15) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0
    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 <= 4.15e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 4.15e+15:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 4.15e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 4.15e+15)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.15e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.15e15

   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 96.4%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-/r* 99.8%

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

       2. *-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around 0 67.9%

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

   if 4.15e15 < beta

   1. Initial program 86.8%

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

   3. Simplified 61.3%

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

      1. times-frac 91.2%

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

      2. +-commutative 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. +-commutative 91.2%

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

      2. +-commutative 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. +-commutative 91.2%

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

      6. +-commutative 91.2%

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

      7. +-commutative 91.2%

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

      8. associate-+r+ 91.2%

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

      9. +-commutative 91.2%

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

      10. +-commutative 91.2%

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

   8. Simplified 91.2%

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

      1. associate-*r/ 91.3%

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

      2. +-commutative 91.3%

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

      3. +-commutative 91.3%

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

      4. *-commutative 91.3%

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

      5. +-commutative 91.3%

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

      6. +-commutative 91.3%

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

      7. associate-+r+ 91.3%

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

   10. Applied egg-rr 91.3%

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

       1. associate-/r* 99.6%

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

       2. *-commutative 99.6%

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

       3. +-commutative 99.6%

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

       4. +-commutative 99.6%

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

       5. +-commutative 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in beta around inf 85.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2e+15)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.2d+15) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2e+15:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2e+15)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2e15

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   7. Simplified 85.5%

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

      1. add-exp-log 85.5%

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

      2. associate-/l/ 85.5%

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

      3. log-div 85.5%

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

      4. log1p-define 85.5%

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

      5. associate-+r+ 85.5%

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

   9. Applied egg-rr 85.5%

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

       1. exp-diff 85.5%

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

       2. log1p-undefine 85.5%

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

       3. rem-exp-log 85.5%

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

       4. rem-exp-log 85.5%

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

       5. *-commutative 85.5%

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

       6. *-commutative 85.5%

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

       7. +-commutative 85.5%

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

       8. +-commutative 85.5%

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

       9. +-commutative 85.5%

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

   11. Simplified 85.5%

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

   12. Taylor expanded in alpha around 0 66.9%

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

   if 2.2e15 < beta

   1. Initial program 86.8%

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

   3. Simplified 61.3%

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

      1. times-frac 91.2%

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

      2. +-commutative 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. +-commutative 91.2%

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

      2. +-commutative 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. +-commutative 91.2%

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

      6. +-commutative 91.2%

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

      7. +-commutative 91.2%

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

      8. associate-+r+ 91.2%

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

      9. +-commutative 91.2%

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

      10. +-commutative 91.2%

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

   8. Simplified 91.2%

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

      1. associate-*r/ 91.3%

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

      2. +-commutative 91.3%

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

      3. +-commutative 91.3%

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

      4. *-commutative 91.3%

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

      5. +-commutative 91.3%

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

      6. +-commutative 91.3%

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

      7. associate-+r+ 91.3%

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

   10. Applied egg-rr 91.3%

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

       1. associate-/r* 99.6%

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

       2. *-commutative 99.6%

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

       3. +-commutative 99.6%

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

       4. +-commutative 99.6%

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

       5. +-commutative 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in beta around inf 85.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.7e+15)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.7e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.7d+15) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.7e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.7e+15:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.7e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.7e+15)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.7e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.7e15

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   7. Simplified 85.5%

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

      1. add-exp-log 85.5%

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

      2. associate-/l/ 85.5%

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

      3. log-div 85.5%

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

      4. log1p-define 85.5%

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

      5. associate-+r+ 85.5%

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

   9. Applied egg-rr 85.5%

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

       1. exp-diff 85.5%

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

       2. log1p-undefine 85.5%

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

       3. rem-exp-log 85.5%

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

       4. rem-exp-log 85.5%

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

       5. *-commutative 85.5%

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

       6. *-commutative 85.5%

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

       7. +-commutative 85.5%

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

       8. +-commutative 85.5%

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

       9. +-commutative 85.5%

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

   11. Simplified 85.5%

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

   12. Taylor expanded in alpha around 0 66.9%

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

   if 5.7e15 < beta

   1. Initial program 86.8%

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

   3. Taylor expanded in beta around inf 84.5%

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

      1. div-inv 84.4%

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

      2. metadata-eval 84.4%

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

      3. associate-+l+ 84.4%

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

      4. metadata-eval 84.4%

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

      5. associate-+r+ 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.5%

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

      2. *-rgt-identity 84.5%

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

      3. +-commutative 84.5%

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

      4. +-commutative 84.5%

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

      5. +-commutative 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (/ (+ 0.5 (* beta 0.25)) (* (+ alpha (+ beta 2.0)) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = (0.5 + (beta * 0.25)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.4d0) then
        tmp = (0.5d0 + (beta * 0.25d0)) / ((alpha + (beta + 2.0d0)) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = (0.5 + (beta * 0.25)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.4:
		tmp = (0.5 + (beta * 0.25)) / ((alpha + (beta + 2.0)) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.4)
		tmp = Float64(Float64(0.5 + Float64(beta * 0.25)) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4)
		tmp = (0.5 + (beta * 0.25)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.4], N[(N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 84.4%

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

      1. *-commutative 84.4%

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

   10. Simplified 84.4%

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

   11. Taylor expanded in alpha around 0 67.1%

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

   if 4.4000000000000004 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 83.7%

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

      1. div-inv 83.6%

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

      2. metadata-eval 83.6%

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

      3. associate-+l+ 83.6%

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

      4. metadata-eval 83.6%

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

      5. associate-+r+ 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. associate-*r/ 83.7%

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

      2. *-rgt-identity 83.7%

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

      3. +-commutative 83.7%

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

      4. +-commutative 83.7%

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

      5. +-commutative 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = (0.5d0 + (beta * 0.25d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 84.4%

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

      1. *-commutative 84.4%

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

   10. Simplified 84.4%

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

   11. Taylor expanded in alpha around 0 66.1%

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

   if 4 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 83.7%

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

      1. div-inv 83.6%

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

      2. metadata-eval 83.6%

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

      3. associate-+l+ 83.6%

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

      4. metadata-eval 83.6%

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

      5. associate-+r+ 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. associate-*r/ 83.7%

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

      2. *-rgt-identity 83.7%

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

      3. +-commutative 83.7%

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

      4. +-commutative 83.7%

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

      5. +-commutative 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 72.6%

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

Alternative 13: 96.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.6d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.6)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.6], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 83.6%

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

   if 7.5999999999999996 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 83.7%

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

      1. div-inv 83.6%

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

      2. metadata-eval 83.6%

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

      3. associate-+l+ 83.6%

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

      4. metadata-eval 83.6%

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

      5. associate-+r+ 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. associate-*r/ 83.7%

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

      2. *-rgt-identity 83.7%

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

      3. +-commutative 83.7%

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

      4. +-commutative 83.7%

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

      5. +-commutative 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.6)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.6d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.6:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.6)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.6)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.6], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.59999999999999964

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 83.6%

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

   if 8.59999999999999964 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 83.7%

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

   4. Taylor expanded in alpha around 0 83.5%

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

      1. +-commutative 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ 1.0 (* beta (/ beta (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.6d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = 1.0d0 / (beta * (beta / (1.0d0 + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta / Float64(1.0 + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.6)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.6], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 83.6%

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

   if 7.5999999999999996 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 83.7%

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

      1. clear-num 83.2%

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

      2. inv-pow 83.2%

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

      3. metadata-eval 83.2%

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

      4. associate-+l+ 83.2%

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

      5. metadata-eval 83.2%

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

      6. associate-+r+ 83.2%

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

   5. Applied egg-rr 83.2%

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

      1. unpow-1 83.2%

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

      2. associate-/r/ 83.3%

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

      3. +-commutative 83.3%

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

      4. +-commutative 83.3%

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

      5. +-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in beta around inf 83.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.4d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.4:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.4)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 83.6%

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

   if 3.39999999999999991 < beta

   1. Initial program 87.1%

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

   3. Taylor expanded in beta around inf 83.7%

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

      1. clear-num 83.2%

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

      2. inv-pow 83.2%

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

      3. metadata-eval 83.2%

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

      4. associate-+l+ 83.2%

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

      5. metadata-eval 83.2%

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

      6. associate-+r+ 83.2%

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

   5. Applied egg-rr 83.2%

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

      1. unpow-1 83.2%

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

      2. associate-/r/ 83.3%

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

      3. +-commutative 83.3%

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

      4. +-commutative 83.3%

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

      5. +-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 76.1%

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

      1. associate-/r* 76.4%

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

   10. Simplified 76.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in beta around inf 32.7%

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

   1. clear-num 32.5%

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

   2. inv-pow 32.5%

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

   3. metadata-eval 32.5%

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

   4. associate-+l+ 32.5%

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

   5. metadata-eval 32.5%

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

   6. associate-+r+ 32.5%

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

5. Applied egg-rr 32.5%

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

   1. unpow-1 32.5%

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

   2. associate-/r/ 32.6%

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

   3. +-commutative 32.6%

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

   4. +-commutative 32.6%

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

   5. +-commutative 32.6%

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

7. Simplified 32.6%

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

8. Taylor expanded in alpha around 0 29.9%

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

   1. associate-/r* 30.1%

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

10. Simplified 30.1%

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

11. Final simplification 30.1%

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

Alternative 18: 50.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in beta around inf 32.7%

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

4. Taylor expanded in alpha around 0 29.9%

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

   1. +-commutative 29.9%

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

6. Simplified 29.9%

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

Alternative 19: 6.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\frac{\beta}{0.25}} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 1.0 (/ beta 0.25)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / (beta / 0.25);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0 / (beta / 0.25d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 1.0 / (beta / 0.25);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 1.0 / (beta / 0.25)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(1.0 / Float64(beta / 0.25))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 1.0 / (beta / 0.25);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(1.0 / N[(beta / 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

   1. associate-/l/ 94.2%

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

   2. +-commutative 94.2%

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

   3. associate-+l+ 94.2%

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

   4. *-commutative 94.2%

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

   5. metadata-eval 94.2%

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

   6. associate-+l+ 94.2%

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

   7. metadata-eval 94.2%

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

   8. +-commutative 94.2%

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

   9. +-commutative 94.2%

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

   10. +-commutative 94.2%

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

   11. metadata-eval 94.2%

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

   12. metadata-eval 94.2%

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

   13. associate-+l+ 94.2%

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

3. Simplified 94.2%

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

5. Taylor expanded in alpha around 0 84.6%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 84.6%

      \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\beta + 2}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

7. Simplified 84.6%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\beta + 2}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

8. Taylor expanded in beta around 0 69.3%

   \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 69.3%

      \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

10. Simplified 69.3%

    \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

11. Taylor expanded in beta around inf 4.4%

    \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
12. Step-by-step derivation

    1. clear-num 4.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta}{0.25}}} \]

    2. inv-pow 4.4%

       \[\leadsto \color{blue}{{\left(\frac{\beta}{0.25}\right)}^{-1}} \]

13. Applied egg-rr 4.4%

    \[\leadsto \color{blue}{{\left(\frac{\beta}{0.25}\right)}^{-1}} \]
14. Step-by-step derivation

    1. unpow-1 4.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta}{0.25}}} \]

15. Simplified 4.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{\beta}{0.25}}} \]
16. Add Preprocessing

Alternative 20: 6.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / 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 = 0.25d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / beta), $MachinePrecision]

Derivation

1. Initial program 95.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 94.2%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative 94.2%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. associate-+l+ 94.2%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. *-commutative 94.2%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. metadata-eval 94.2%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. associate-+l+ 94.2%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. metadata-eval 94.2%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. +-commutative 94.2%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 94.2%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(2 + \color{blue}{\left(\beta + \alpha\right)}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. +-commutative 94.2%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 2\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. metadata-eval 94.2%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. metadata-eval 94.2%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

   13. associate-+l+ 94.2%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

3. Simplified 94.2%

   \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in alpha around 0 84.6%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 84.6%

      \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\beta + 2}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

7. Simplified 84.6%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\beta + 2}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

8. Taylor expanded in beta around 0 69.3%

   \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 69.3%

      \[\leadsto \frac{0.5 + \color{blue}{\beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

10. Simplified 69.3%

    \[\leadsto \frac{\color{blue}{0.5 + \beta \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

11. Taylor expanded in beta around inf 4.4%

    \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))