Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 23.3s

Alternatives: 19

Speedup: 1.9×

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: 94.4% 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, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(3 + \beta\right)\\ t_1 := 2 + \left(\alpha + \beta\right)\\ t_2 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 6 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \alpha}{t_2} \cdot \frac{1 + \beta}{t_0 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{t_1} \cdot \frac{\beta}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 3.0 beta)))
        (t_1 (+ 2.0 (+ alpha beta)))
        (t_2 (+ alpha (+ beta 2.0))))
   (if (<= beta 6e+16)
     (* (/ (+ 1.0 alpha) t_2) (/ (+ 1.0 beta) (* t_0 t_2)))
     (* (/ (/ (+ 1.0 alpha) t_0) t_1) (/ beta t_1)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (3.0 + beta);
	double t_1 = 2.0 + (alpha + beta);
	double t_2 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6e+16) {
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_0 * t_2));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_1) * (beta / t_1);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = alpha + (3.0d0 + beta)
    t_1 = 2.0d0 + (alpha + beta)
    t_2 = alpha + (beta + 2.0d0)
    if (beta <= 6d+16) then
        tmp = ((1.0d0 + alpha) / t_2) * ((1.0d0 + beta) / (t_0 * t_2))
    else
        tmp = (((1.0d0 + alpha) / t_0) / t_1) * (beta / t_1)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (3.0 + beta);
	double t_1 = 2.0 + (alpha + beta);
	double t_2 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6e+16) {
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_0 * t_2));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_1) * (beta / t_1);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (3.0 + beta)
	t_1 = 2.0 + (alpha + beta)
	t_2 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 6e+16:
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_0 * t_2))
	else:
		tmp = (((1.0 + alpha) / t_0) / t_1) * (beta / t_1)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(3.0 + beta))
	t_1 = Float64(2.0 + Float64(alpha + beta))
	t_2 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 6e+16)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_2) * Float64(Float64(1.0 + beta) / Float64(t_0 * t_2)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_1) * Float64(beta / t_1));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (3.0 + beta);
	t_1 = 2.0 + (alpha + beta);
	t_2 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 6e+16)
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_0 * t_2));
	else
		tmp = (((1.0 + alpha) / t_0) / t_1) * (beta / t_1);
	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[(3.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6e+16], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(beta / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6e16

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   if 6e16 < beta

   1. Initial program 76.6%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in beta around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. *-commutative 48.3%

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

      3. times-frac 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-+r+ 84.7%

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

      6. +-commutative 84.7%

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

      7. +-commutative 84.7%

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

      8. associate-+r+ 84.7%

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

      9. +-commutative 84.7%

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

      10. +-commutative 84.7%

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

   7. Applied egg-rr 84.7%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(3 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{2 + \left(\alpha + \beta\right)} \cdot \frac{\beta}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + \beta}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\alpha + \left(3 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.7e+45)
   (*
    (/ (+ 1.0 beta) (+ 2.0 (+ alpha beta)))
    (/ (/ 1.0 (+ beta 2.0)) (+ 3.0 beta)))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ alpha (+ 3.0 beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7e+45) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (3.0 + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.7d+45) then
        tmp = ((1.0d0 + beta) / (2.0d0 + (alpha + beta))) * ((1.0d0 / (beta + 2.0d0)) / (3.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (alpha + (3.0d0 + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7e+45) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (3.0 + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.7e+45:
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (3.0 + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.7e+45)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + Float64(alpha + beta))) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(alpha + Float64(3.0 + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.7e+45)
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + ((-1.0 - alpha) / beta)) / (alpha + (3.0 + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.7e+45], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $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[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.69999999999999977e45

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. associate-/l/ 99.4%

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

       2. *-commutative 99.4%

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

       3. +-commutative 99.4%

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

       4. times-frac 99.8%

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

   11. Applied egg-rr 99.4%

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

       1. associate-/r* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in alpha around 0 66.8%

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

       1. associate-/r* 66.8%

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

       2. +-commutative 66.8%

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

   16. Simplified 66.8%

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

   if 3.69999999999999977e45 < beta

   1. Initial program 75.2%

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

   3. Simplified 83.8%

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

      1. clear-num 83.8%

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

      2. associate-+r+ 83.8%

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

      3. *-commutative 83.8%

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

      4. frac-times 65.1%

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

      5. *-un-lft-identity 65.1%

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

      6. +-commutative 65.1%

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

      7. *-commutative 65.1%

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

      8. associate-+r+ 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. associate-/r* 83.8%

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

      2. associate-/l* 68.0%

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

      3. associate-*l/ 83.8%

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

      4. *-commutative 83.8%

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

      5. times-frac 99.6%

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

      6. associate-/r* 83.8%

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

      7. *-commutative 83.8%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 71.7%

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

       1. mul-1-neg 71.7%

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

       2. unsub-neg 71.7%

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

   11. Simplified 71.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + \beta}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{\alpha + \left(3 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 19500000000000:\\ \;\;\;\;\frac{1 + \beta}{t_0} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{t_0} \cdot \frac{\beta}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 19500000000000.0)
     (* (/ (+ 1.0 beta) t_0) (/ (/ 1.0 (+ beta 2.0)) (+ 3.0 beta)))
     (* (/ (/ (+ 1.0 alpha) (+ alpha (+ 3.0 beta))) t_0) (/ beta t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 19500000000000.0) {
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0) * (beta / t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 19500000000000.0d0) then
        tmp = ((1.0d0 + beta) / t_0) * ((1.0d0 / (beta + 2.0d0)) / (3.0d0 + beta))
    else
        tmp = (((1.0d0 + alpha) / (alpha + (3.0d0 + beta))) / t_0) * (beta / t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 19500000000000.0) {
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0) * (beta / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 19500000000000.0:
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta))
	else:
		tmp = (((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0) * (beta / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 19500000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(3.0 + beta))) / t_0) * Float64(beta / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 19500000000000.0)
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	else
		tmp = (((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0) * (beta / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 19500000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.95e13

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. associate-/l/ 99.4%

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

       2. *-commutative 99.4%

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

       3. +-commutative 99.4%

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

       4. times-frac 99.8%

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

   11. Applied egg-rr 99.4%

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

       1. associate-/r* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in alpha around 0 67.6%

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

       1. associate-/r* 67.6%

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

       2. +-commutative 67.6%

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

   16. Simplified 67.6%

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

   if 1.95e13 < beta

   1. Initial program 76.6%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in beta around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. *-commutative 48.3%

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

      3. times-frac 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-+r+ 84.7%

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

      6. +-commutative 84.7%

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

      7. +-commutative 84.7%

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

      8. associate-+r+ 84.7%

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

      9. +-commutative 84.7%

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

      10. +-commutative 84.7%

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

   7. Applied egg-rr 84.7%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 19500000000000:\\ \;\;\;\;\frac{1 + \beta}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{2 + \left(\alpha + \beta\right)} \cdot \frac{\beta}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.66 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(3 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.66e+16)
   (*
    (/ (+ 1.0 beta) (+ 2.0 (+ alpha beta)))
    (/ (/ 1.0 (+ beta 2.0)) (+ 3.0 beta)))
   (/
    (/ (* (+ 1.0 beta) (/ (+ 1.0 alpha) beta)) (+ alpha (+ beta 2.0)))
    (+ alpha (+ 3.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.66e+16) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = (((1.0 + beta) * ((1.0 + alpha) / beta)) / (alpha + (beta + 2.0))) / (alpha + (3.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.66d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + (alpha + beta))) * ((1.0d0 / (beta + 2.0d0)) / (3.0d0 + beta))
    else
        tmp = (((1.0d0 + beta) * ((1.0d0 + alpha) / beta)) / (alpha + (beta + 2.0d0))) / (alpha + (3.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.66e+16) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = (((1.0 + beta) * ((1.0 + alpha) / beta)) / (alpha + (beta + 2.0))) / (alpha + (3.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.66e+16:
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta))
	else:
		tmp = (((1.0 + beta) * ((1.0 + alpha) / beta)) / (alpha + (beta + 2.0))) / (alpha + (3.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.66e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + Float64(alpha + beta))) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(Float64(1.0 + alpha) / beta)) / Float64(alpha + Float64(beta + 2.0))) / Float64(alpha + Float64(3.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.66e+16)
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	else
		tmp = (((1.0 + beta) * ((1.0 + alpha) / beta)) / (alpha + (beta + 2.0))) / (alpha + (3.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.66e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.66e16

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. associate-/l/ 99.4%

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

       2. *-commutative 99.4%

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

       3. +-commutative 99.4%

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

       4. times-frac 99.8%

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

   11. Applied egg-rr 99.4%

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

       1. associate-/r* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in alpha around 0 67.6%

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

       1. associate-/r* 67.6%

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

       2. +-commutative 67.6%

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

   16. Simplified 67.6%

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

   if 1.66e16 < beta

   1. Initial program 76.6%

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

      1. div-inv 76.5%

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

      2. +-commutative 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-+r+ 76.5%

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

      5. +-commutative 76.5%

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

      6. distribute-rgt1-in 76.5%

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

      7. fma-def 76.5%

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

      8. +-commutative 76.5%

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

      9. metadata-eval 76.5%

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

      10. associate-+r+ 76.5%

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

      11. metadata-eval 76.5%

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

      12. associate-+r+ 76.5%

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

   4. Applied egg-rr 76.5%

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

      1. associate-*l/ 76.5%

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

      2. associate-*r/ 76.6%

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

      3. fma-udef 76.6%

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

      4. *-commutative 76.6%

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

      5. +-commutative 76.6%

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

      6. associate-+r+ 76.6%

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

      7. +-commutative 76.6%

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

      8. distribute-lft1-in 76.6%

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

      9. *-rgt-identity 76.6%

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

      10. +-commutative 76.6%

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

      11. *-commutative 76.6%

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

      12. +-commutative 76.6%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in beta around inf 70.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.66 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(3 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.4× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. div-inv 93.3%

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

   2. +-commutative 93.3%

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

   3. *-commutative 93.3%

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

   4. associate-+r+ 93.3%

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

   5. +-commutative 93.3%

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

   6. distribute-rgt1-in 93.3%

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

   7. fma-def 93.3%

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

   8. +-commutative 93.3%

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

   9. metadata-eval 93.3%

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

   10. associate-+r+ 93.3%

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

   11. metadata-eval 93.3%

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

   12. associate-+r+ 93.3%

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

4. Applied egg-rr 93.3%

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

   1. associate-*l/ 93.3%

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

   2. associate-*r/ 93.3%

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

   3. fma-udef 93.3%

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

   4. *-commutative 93.3%

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

   5. +-commutative 93.3%

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

   6. associate-+r+ 93.3%

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

   7. +-commutative 93.3%

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

   8. distribute-lft1-in 93.3%

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

   9. *-rgt-identity 93.3%

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

   10. +-commutative 93.3%

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

   11. *-commutative 93.3%

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

   12. +-commutative 93.3%

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

   13. associate-*l/ 99.8%

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

   14. +-commutative 99.8%

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

   15. *-commutative 99.8%

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

   16. +-commutative 99.8%

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

   17. +-commutative 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in alpha around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

9. Simplified 99.8%

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

    1. associate-/l/ 95.3%

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

    2. *-commutative 95.3%

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

    3. +-commutative 95.3%

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

    4. times-frac 99.8%

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

11. Applied egg-rr 95.3%

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

    1. associate-/r* 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

13. Simplified 99.8%

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

    1. associate-*r/ 99.8%

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

    2. +-commutative 99.8%

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

    3. associate-+r+ 99.8%

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

    4. +-commutative 99.8%

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

    5. associate-+r+ 99.8%

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

15. Applied egg-rr 99.8%

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

16. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. div-inv 93.3%

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

   2. +-commutative 93.3%

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

   3. *-commutative 93.3%

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

   4. associate-+r+ 93.3%

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

   5. +-commutative 93.3%

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

   6. distribute-rgt1-in 93.3%

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

   7. fma-def 93.3%

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

   8. +-commutative 93.3%

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

   9. metadata-eval 93.3%

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

   10. associate-+r+ 93.3%

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

   11. metadata-eval 93.3%

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

   12. associate-+r+ 93.3%

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

4. Applied egg-rr 93.3%

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

   1. associate-*l/ 93.3%

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

   2. associate-*r/ 93.3%

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

   3. fma-udef 93.3%

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

   4. *-commutative 93.3%

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

   5. +-commutative 93.3%

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

   6. associate-+r+ 93.3%

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

   7. +-commutative 93.3%

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

   8. distribute-lft1-in 93.3%

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

   9. *-rgt-identity 93.3%

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

   10. +-commutative 93.3%

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

   11. *-commutative 93.3%

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

   12. +-commutative 93.3%

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

   13. associate-*l/ 99.8%

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

   14. +-commutative 99.8%

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

   15. *-commutative 99.8%

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

   16. +-commutative 99.8%

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

   17. +-commutative 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in alpha around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

9. Simplified 99.8%

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

    1. associate-/l/ 95.3%

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

    2. *-commutative 95.3%

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

    3. +-commutative 95.3%

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

    4. times-frac 99.8%

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

11. Applied egg-rr 95.3%

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

    1. associate-/r* 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 7: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \beta}{t_0} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 2.65e+18)
     (* (/ (+ 1.0 beta) t_0) (/ (/ 1.0 (+ beta 2.0)) (+ 3.0 beta)))
     (/ (/ (+ 1.0 alpha) (+ alpha (+ 3.0 beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.65e+18) {
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 2.65d+18) then
        tmp = ((1.0d0 + beta) / t_0) * ((1.0d0 / (beta + 2.0d0)) / (3.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (3.0d0 + beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.65e+18) {
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 2.65e+18:
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta))
	else:
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2.65e+18)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(3.0 + beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 2.65e+18)
		tmp = ((1.0 + beta) / t_0) * ((1.0 / (beta + 2.0)) / (3.0 + beta));
	else
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.65e+18], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.65e18

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. associate-/l/ 99.4%

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

       2. *-commutative 99.4%

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

       3. +-commutative 99.4%

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

       4. times-frac 99.8%

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

   11. Applied egg-rr 99.4%

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

       1. associate-/r* 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in alpha around 0 67.3%

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

       1. associate-/r* 67.3%

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

       2. +-commutative 67.3%

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

   16. Simplified 67.3%

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

   if 2.65e18 < beta

   1. Initial program 76.2%

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

      1. div-inv 76.2%

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

      2. +-commutative 76.2%

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

      3. *-commutative 76.2%

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

      4. associate-+r+ 76.2%

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

      5. +-commutative 76.2%

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

      6. distribute-rgt1-in 76.2%

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

      7. fma-def 76.2%

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

      8. +-commutative 76.2%

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

      9. metadata-eval 76.2%

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

      10. associate-+r+ 76.2%

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

      11. metadata-eval 76.2%

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

      12. associate-+r+ 76.2%

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

   4. Applied egg-rr 76.2%

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

      1. associate-*l/ 76.2%

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

      2. associate-*r/ 76.2%

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

      3. fma-udef 76.2%

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

      4. *-commutative 76.2%

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

      5. +-commutative 76.2%

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

      6. associate-+r+ 76.2%

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

      7. +-commutative 76.2%

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

      8. distribute-lft1-in 76.2%

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

      9. *-rgt-identity 76.2%

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

      10. +-commutative 76.2%

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

      11. *-commutative 76.2%

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

      12. +-commutative 76.2%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around inf 71.6%

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

      1. expm1-log1p-u 71.6%

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

      2. expm1-udef 47.9%

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

      3. associate-/l/ 47.9%

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

      4. metadata-eval 47.9%

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

      5. associate-+l+ 47.9%

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

      6. metadata-eval 47.9%

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

      7. associate-+r+ 47.9%

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

      8. +-commutative 47.9%

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

      9. associate-+r+ 47.9%

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

      10. +-commutative 47.9%

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

      11. +-commutative 47.9%

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

   9. Applied egg-rr 47.9%

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

       1. expm1-def 79.4%

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

       2. expm1-log1p 79.4%

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

       3. associate-/r* 71.6%

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

       4. +-commutative 71.6%

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

       5. +-commutative 71.6%

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

   11. Simplified 71.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \beta}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\beta + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(3 + \beta\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+14)
   (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (* (+ 3.0 beta) (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 3.0 beta))) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+14) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((3.0 + beta) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+14) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * ((3.0d0 + beta) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (3.0d0 + beta))) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+14) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((3.0 + beta) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5e+14:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((3.0 + beta) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+14)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(3.0 + beta) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(3.0 + beta))) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+14)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((3.0 + beta) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5e+14], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(3.0 + beta), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e14

   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 94.9%

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

   5. Taylor expanded in alpha around 0 84.1%

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

   6. Taylor expanded in alpha around 0 67.6%

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

   if 5e14 < beta

   1. Initial program 76.6%

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

      1. div-inv 76.5%

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

      2. +-commutative 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-+r+ 76.5%

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

      5. +-commutative 76.5%

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

      6. distribute-rgt1-in 76.5%

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

      7. fma-def 76.5%

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

      8. +-commutative 76.5%

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

      9. metadata-eval 76.5%

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

      10. associate-+r+ 76.5%

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

      11. metadata-eval 76.5%

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

      12. associate-+r+ 76.5%

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

   4. Applied egg-rr 76.5%

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

      1. associate-*l/ 76.5%

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

      2. associate-*r/ 76.6%

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

      3. fma-udef 76.6%

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

      4. *-commutative 76.6%

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

      5. +-commutative 76.6%

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

      6. associate-+r+ 76.6%

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

      7. +-commutative 76.6%

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

      8. distribute-lft1-in 76.6%

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

      9. *-rgt-identity 76.6%

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

      10. +-commutative 76.6%

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

      11. *-commutative 76.6%

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

      12. +-commutative 76.6%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around inf 70.7%

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

      1. expm1-log1p-u 70.7%

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

      2. expm1-udef 47.4%

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

      3. associate-/l/ 47.4%

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

      4. metadata-eval 47.4%

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

      5. associate-+l+ 47.4%

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

      6. metadata-eval 47.4%

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

      7. associate-+r+ 47.4%

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

      8. +-commutative 47.4%

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

      9. associate-+r+ 47.4%

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

      10. +-commutative 47.4%

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

      11. +-commutative 47.4%

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

   9. Applied egg-rr 47.4%

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

       1. expm1-def 78.3%

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

       2. expm1-log1p 78.3%

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

       3. associate-/r* 70.7%

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

       4. +-commutative 70.7%

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

       5. +-commutative 70.7%

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

   11. Simplified 70.7%

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

4. Final simplification 68.5%

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

Alternative 9: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (+ 6.0 (* beta 5.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 3.0 beta))) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * (6.0d0 + (beta * 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (3.0d0 + beta))) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(6.0 + Float64(beta * 5.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(3.0 + beta))) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (3.0 + beta))) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in beta around 0 99.1%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\color{blue}{\beta \cdot \left(5 + 2 \cdot \alpha\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]

   6. Taylor expanded in alpha around 0 65.9%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + 5 \cdot \beta\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \color{blue}{\beta \cdot 5}\right)} \]

   8. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot 5\right)}} \]

   if 2.10000000000000009 < beta

   1. Initial program 76.9%

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

      1. div-inv 76.8%

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

      2. +-commutative 76.8%

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

      3. *-commutative 76.8%

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

      4. associate-+r+ 76.8%

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

      5. +-commutative 76.8%

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

      6. distribute-rgt1-in 76.8%

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

      7. fma-def 76.8%

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

      8. +-commutative 76.8%

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

      9. metadata-eval 76.8%

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

      10. associate-+r+ 76.8%

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

      11. metadata-eval 76.8%

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

      12. associate-+r+ 76.8%

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

   4. Applied egg-rr 76.8%

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

      1. associate-*l/ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. fma-udef 76.9%

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

      4. *-commutative 76.9%

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

      5. +-commutative 76.9%

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

      6. associate-+r+ 76.9%

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

      7. +-commutative 76.9%

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

      8. distribute-lft1-in 76.9%

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

      9. *-rgt-identity 76.9%

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

      10. +-commutative 76.9%

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

      11. *-commutative 76.9%

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

      12. +-commutative 76.9%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around inf 70.3%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 47.3%

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

      3. associate-/l/ 47.3%

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

      4. metadata-eval 47.3%

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

      5. associate-+l+ 47.3%

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

      6. metadata-eval 47.3%

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

      7. associate-+r+ 47.3%

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

      8. +-commutative 47.3%

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

      9. associate-+r+ 47.3%

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

      10. +-commutative 47.3%

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

      11. +-commutative 47.3%

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

   9. Applied egg-rr 47.3%

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

       1. expm1-def 77.9%

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

       2. expm1-log1p 77.9%

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

       3. associate-/r* 70.3%

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

       4. +-commutative 70.3%

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

       5. +-commutative 70.3%

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

   11. Simplified 70.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(3 + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (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 <= 5.0d0) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * (6.0d0 + (beta * 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (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 <= 5.0) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in beta around 0 99.1%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\color{blue}{\beta \cdot \left(5 + 2 \cdot \alpha\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]

   6. Taylor expanded in alpha around 0 65.9%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + 5 \cdot \beta\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \color{blue}{\beta \cdot 5}\right)} \]

   8. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot 5\right)}} \]

   if 5 < beta

   1. Initial program 76.9%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in beta around inf 69.3%

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

      1. associate-*l/ 69.4%

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

      2. +-commutative 69.4%

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

   7. Applied egg-rr 69.4%

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

      1. associate-*r/ 69.4%

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

      2. *-rgt-identity 69.4%

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

      3. +-commutative 69.4%

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

   9. Simplified 69.4%

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

4. Final simplification 66.9%

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

Alternative 11: 62.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(3 + \beta\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5e+16)
   (/ 1.0 (* (+ 3.0 beta) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5e+16) {
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (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 <= 3.5d+16) then
        tmp = 1.0d0 / ((3.0d0 + beta) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (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 <= 3.5e+16) {
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.5e+16:
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5e+16)
		tmp = Float64(1.0 / Float64(Float64(3.0 + beta) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / 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 <= 3.5e+16)
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (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, 3.5e+16], N[(1.0 / N[(N[(3.0 + beta), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5e16

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around inf 15.1%

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

   8. Taylor expanded in alpha around 0 14.1%

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

   if 3.5e16 < beta

   1. Initial program 76.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in beta around inf 69.7%

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

      1. associate-*l/ 69.8%

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

      2. +-commutative 69.8%

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

   7. Applied egg-rr 69.8%

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

      1. associate-*r/ 69.8%

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

      2. *-rgt-identity 69.8%

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

      3. +-commutative 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(3 + \beta\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.15)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.15d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.15:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.15)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.15)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.15], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1499999999999999

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around inf 14.9%

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

   8. Taylor expanded in alpha around 0 13.9%

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

      1. associate-/r* 13.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   11. Taylor expanded in beta around 0 13.9%

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

       1. *-commutative 13.9%

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

   13. Simplified 13.9%

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

   if 1.1499999999999999 < beta

   1. Initial program 76.9%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in beta around inf 69.3%

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

   6. Taylor expanded in beta around inf 68.9%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\left(3 + \beta\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.2e+38)
   (/ 1.0 (* (+ 3.0 beta) (+ beta 2.0)))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2e+38) {
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.2d+38) then
        tmp = 1.0d0 / ((3.0d0 + beta) * (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2e+38) {
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.2e+38:
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.2e+38)
		tmp = Float64(1.0 / Float64(Float64(3.0 + beta) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.2e+38)
		tmp = 1.0 / ((3.0 + beta) * (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.2e+38], N[(1.0 / N[(N[(3.0 + beta), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.19999999999999938e38

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around inf 15.4%

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

   8. Taylor expanded in alpha around 0 14.4%

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

   if 7.19999999999999938e38 < beta

   1. Initial program 75.2%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in beta around inf 72.1%

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

   6. Taylor expanded in beta around inf 71.8%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\left(3 + \beta\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.12:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(3 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.12)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ 1.0 (* beta (+ 3.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.12) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / (beta * (3.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.12d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = 1.0d0 / (beta * (3.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.12) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / (beta * (3.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.12:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = 1.0 / (beta * (3.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.12)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(3.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.12)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = 1.0 / (beta * (3.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.12], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1200000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

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

      14. +-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around inf 14.9%

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

   8. Taylor expanded in alpha around 0 13.9%

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

      1. associate-/r* 13.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   11. Taylor expanded in beta around 0 13.9%

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

       1. *-commutative 13.9%

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

   13. Simplified 13.9%

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

   if 1.1200000000000001 < beta

   1. Initial program 76.9%

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

   3. Taylor expanded in beta around -inf 69.4%

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

   4. Taylor expanded in alpha around 0 66.4%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.12:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(3 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.12:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.12)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ (/ 1.0 beta) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.12) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = (1.0 / beta) / (3.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.12d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = (1.0d0 / beta) / (3.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.12) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = (1.0 / beta) / (3.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.12:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = (1.0 / beta) / (3.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.12)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(3.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.12)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = (1.0 / beta) / (3.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.12], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1200000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-+r+ 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft1-in 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-*l/ 99.8%

          \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      15. *-commutative 99.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      16. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      17. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Taylor expanded in beta around inf 14.9%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Taylor expanded in alpha around 0 13.9%

      \[\leadsto \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 13.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   11. Taylor expanded in beta around 0 13.9%

       \[\leadsto \color{blue}{0.16666666666666666 + -0.1388888888888889 \cdot \beta} \]
   12. Step-by-step derivation

       1. *-commutative 13.9%

          \[\leadsto 0.16666666666666666 + \color{blue}{\beta \cdot -0.1388888888888889} \]

   13. Simplified 13.9%

       \[\leadsto \color{blue}{0.16666666666666666 + \beta \cdot -0.1388888888888889} \]

   if 1.1200000000000001 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around -inf 69.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \alpha - 1}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in alpha around 0 66.4%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 66.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{3 + \beta}} \]

      2. +-commutative 66.4%

         \[\leadsto \frac{\frac{1}{\beta}}{\color{blue}{\beta + 3}} \]

   6. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{\beta + 3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.12:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{3 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 12.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.15)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.15d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.15:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.15)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.15)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.15], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1499999999999999

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-commutative 99.8%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. *-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \color{blue}{\alpha \cdot \beta}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\left(\beta + \alpha \cdot \beta\right) + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. distribute-rgt1-in 99.8%

         \[\leadsto \frac{\frac{1 + \left(\color{blue}{\left(\alpha + 1\right) \cdot \beta} + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. fma-def 99.8%

         \[\leadsto \frac{\frac{1 + \color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(\color{blue}{1 + \alpha}, \beta, \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      9. metadata-eval 99.8%

         \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\left(\alpha + \beta\right) + \color{blue}{2}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      11. metadata-eval 99.8%

          \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      12. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   5. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. associate-*r/ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. fma-udef 99.8%

         \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\left(1 + \alpha\right) \cdot \beta + \alpha\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-commutative 99.8%

         \[\leadsto \frac{\frac{\frac{\left(1 + \left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(1 + \alpha\right) + \beta \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \left(1 + \alpha\right) + \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. distribute-lft1-in 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\beta + 1\right) \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      9. *-rgt-identity 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      11. *-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      12. +-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{\left(1 + \alpha\right) \cdot \color{blue}{\left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      13. associate-*l/ 99.8%

          \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      15. *-commutative 99.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      16. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      17. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Taylor expanded in beta around inf 14.9%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Taylor expanded in alpha around 0 13.9%

      \[\leadsto \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 13.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   11. Taylor expanded in beta around 0 13.9%

       \[\leadsto \color{blue}{0.16666666666666666 + -0.1388888888888889 \cdot \beta} \]
   12. Step-by-step derivation

       1. *-commutative 13.9%

          \[\leadsto 0.16666666666666666 + \color{blue}{\beta \cdot -0.1388888888888889} \]

   13. Simplified 13.9%

       \[\leadsto \color{blue}{0.16666666666666666 + \beta \cdot -0.1388888888888889} \]

   if 1.1499999999999999 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 69.3%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]

   6. Taylor expanded in alpha around inf 6.2%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 12.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) 0.16666666666666666 (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.16666666666666666d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.16666666666666666
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = 0.16666666666666666;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], 0.16666666666666666, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.8%

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-commutative 99.8%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. *-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \color{blue}{\alpha \cdot \beta}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\left(\beta + \alpha \cdot \beta\right) + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. distribute-rgt1-in 99.8%

         \[\leadsto \frac{\frac{1 + \left(\color{blue}{\left(\alpha + 1\right) \cdot \beta} + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. fma-def 99.8%

         \[\leadsto \frac{\frac{1 + \color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(\color{blue}{1 + \alpha}, \beta, \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      9. metadata-eval 99.8%

         \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\left(\alpha + \beta\right) + \color{blue}{2}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      11. metadata-eval 99.8%

          \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      12. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   5. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. associate-*r/ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. fma-udef 99.8%

         \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\left(1 + \alpha\right) \cdot \beta + \alpha\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-commutative 99.8%

         \[\leadsto \frac{\frac{\frac{\left(1 + \left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(1 + \alpha\right) + \beta \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \left(1 + \alpha\right) + \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. distribute-lft1-in 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\beta + 1\right) \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      9. *-rgt-identity 99.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      11. *-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      12. +-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{\left(1 + \alpha\right) \cdot \color{blue}{\left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      13. associate-*l/ 99.8%

          \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      15. *-commutative 99.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      16. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      17. +-commutative 99.8%

          \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Taylor expanded in beta around inf 14.9%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Taylor expanded in alpha around 0 13.9%

      \[\leadsto \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 13.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   11. Taylor expanded in beta around 0 13.9%

       \[\leadsto \color{blue}{0.16666666666666666} \]

   if 6 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 69.3%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]

   6. Taylor expanded in alpha around inf 6.2%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 12.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{3 + \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.5 (+ 3.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / (3.0 + beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0 / (3.0d0 + beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / (3.0 + beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / (3.0 + beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / Float64(3.0 + beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / (3.0 + beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.5 / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 93.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   2. +-commutative 93.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. *-commutative 93.3%

      \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \color{blue}{\alpha \cdot \beta}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. associate-+r+ 93.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. +-commutative 93.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\left(\beta + \alpha \cdot \beta\right) + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. distribute-rgt1-in 93.3%

      \[\leadsto \frac{\frac{1 + \left(\color{blue}{\left(\alpha + 1\right) \cdot \beta} + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. fma-def 93.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. +-commutative 93.3%

      \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(\color{blue}{1 + \alpha}, \beta, \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. metadata-eval 93.3%

      \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\left(\alpha + \beta\right) + \color{blue}{2}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   10. associate-+r+ 93.3%

       \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   11. metadata-eval 93.3%

       \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   12. associate-+r+ 93.3%

       \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

4. Applied egg-rr 93.3%

   \[\leadsto \frac{\color{blue}{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
5. Step-by-step derivation

   1. associate-*l/ 93.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   2. associate-*r/ 93.3%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. fma-udef 93.3%

      \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\left(1 + \alpha\right) \cdot \beta + \alpha\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. *-commutative 93.3%

      \[\leadsto \frac{\frac{\frac{\left(1 + \left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. +-commutative 93.3%

      \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. associate-+r+ 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(1 + \alpha\right) + \beta \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. +-commutative 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \left(1 + \alpha\right) + \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. distribute-lft1-in 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\beta + 1\right) \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. *-rgt-identity 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   10. +-commutative 93.3%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   11. *-commutative 93.3%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   12. +-commutative 93.3%

       \[\leadsto \frac{\frac{\frac{\left(1 + \alpha\right) \cdot \color{blue}{\left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   13. associate-*l/ 99.8%

       \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   14. +-commutative 99.8%

       \[\leadsto \frac{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   15. *-commutative 99.8%

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   16. +-commutative 99.8%

       \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   17. +-commutative 99.8%

       \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

6. Simplified 99.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

7. Taylor expanded in beta around inf 30.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

8. Taylor expanded in alpha around 0 28.9%

   \[\leadsto \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
9. Step-by-step derivation

   1. associate-/r* 28.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

10. Simplified 28.9%

    \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

11. Taylor expanded in beta around 0 11.7%

    \[\leadsto \frac{\color{blue}{0.5}}{3 + \beta} \]

12. Final simplification 11.7%

    \[\leadsto \frac{0.5}{3 + \beta} \]
13. Add Preprocessing

Alternative 19: 10.6% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.16666666666666666 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.16666666666666666)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.16666666666666666
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.16666666666666666

Derivation

1. Initial program 93.3%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 93.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   2. +-commutative 93.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. *-commutative 93.3%

      \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \color{blue}{\alpha \cdot \beta}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. associate-+r+ 93.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. +-commutative 93.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\left(\beta + \alpha \cdot \beta\right) + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. distribute-rgt1-in 93.3%

      \[\leadsto \frac{\frac{1 + \left(\color{blue}{\left(\alpha + 1\right) \cdot \beta} + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. fma-def 93.3%

      \[\leadsto \frac{\frac{1 + \color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. +-commutative 93.3%

      \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(\color{blue}{1 + \alpha}, \beta, \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. metadata-eval 93.3%

      \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\left(\alpha + \beta\right) + \color{blue}{2}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   10. associate-+r+ 93.3%

       \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   11. metadata-eval 93.3%

       \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   12. associate-+r+ 93.3%

       \[\leadsto \frac{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

4. Applied egg-rr 93.3%

   \[\leadsto \frac{\color{blue}{\frac{1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
5. Step-by-step derivation

   1. associate-*l/ 93.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   2. associate-*r/ 93.3%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(1 + \mathsf{fma}\left(1 + \alpha, \beta, \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. fma-udef 93.3%

      \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\left(1 + \alpha\right) \cdot \beta + \alpha\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. *-commutative 93.3%

      \[\leadsto \frac{\frac{\frac{\left(1 + \left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right)\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. +-commutative 93.3%

      \[\leadsto \frac{\frac{\frac{\left(1 + \color{blue}{\left(\alpha + \beta \cdot \left(1 + \alpha\right)\right)}\right) \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. associate-+r+ 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(1 + \alpha\right) + \beta \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. +-commutative 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \left(1 + \alpha\right) + \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. distribute-lft1-in 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\beta + 1\right) \cdot \left(1 + \alpha\right)\right)} \cdot 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. *-rgt-identity 93.3%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   10. +-commutative 93.3%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   11. *-commutative 93.3%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   12. +-commutative 93.3%

       \[\leadsto \frac{\frac{\frac{\left(1 + \alpha\right) \cdot \color{blue}{\left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   13. associate-*l/ 99.8%

       \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(\beta + 1\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   14. +-commutative 99.8%

       \[\leadsto \frac{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(1 + \beta\right)}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   15. *-commutative 99.8%

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   16. +-commutative 99.8%

       \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   17. +-commutative 99.8%

       \[\leadsto \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

6. Simplified 99.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

7. Taylor expanded in beta around inf 30.7%

   \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

8. Taylor expanded in alpha around 0 28.9%

   \[\leadsto \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
9. Step-by-step derivation

   1. associate-/r* 28.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

10. Simplified 28.9%

    \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

11. Taylor expanded in beta around 0 11.0%

    \[\leadsto \color{blue}{0.16666666666666666} \]

12. Final simplification 11.0%

    \[\leadsto 0.16666666666666666 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))