Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.5%

Time: 20.6s

Alternatives: 20

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{3 + \left(\alpha + \beta\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+144)
     (/ (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (+ 3.0 (+ alpha beta))) (* t_0 t_0))
     (/ (/ 1.0 beta) (/ beta (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+144) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (3.0 + (alpha + beta))) / (t_0 * t_0);
	} else {
		tmp = (1.0 / beta) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5d+144) then
        tmp = (((1.0d0 + beta) * (1.0d0 + alpha)) / (3.0d0 + (alpha + beta))) / (t_0 * t_0)
    else
        tmp = (1.0d0 / beta) / (beta / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+144) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (3.0 + (alpha + beta))) / (t_0 * t_0);
	} else {
		tmp = (1.0 / beta) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+144:
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (3.0 + (alpha + beta))) / (t_0 * t_0)
	else:
		tmp = (1.0 / beta) / (beta / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+144)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(3.0 + Float64(alpha + beta))) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+144)
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (3.0 + (alpha + beta))) / (t_0 * t_0);
	else
		tmp = (1.0 / beta) / (beta / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+144], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999999e144

   1. Initial program 99.3%

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

      1. associate-/l/ 99.2%

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

      2. associate-/l/ 91.7%

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

      3. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

   if 4.9999999999999999e144 < beta

   1. Initial program 89.5%

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

      1. associate-/l/ 87.4%

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

      2. associate-/l/ 87.4%

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

      3. +-commutative 87.4%

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

      4. *-commutative 87.4%

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

      5. +-commutative 87.4%

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

      6. associate-+l+ 87.4%

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

      7. associate-+l+ 87.4%

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

      8. fma-def 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in beta around inf 95.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 95.1%

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

   6. Simplified 95.1%

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

      1. +-commutative 95.1%

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

      2. associate-/r* 93.9%

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

      3. div-inv 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. clear-num 93.9%

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

      2. associate-*l/ 93.9%

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

      3. div-inv 93.9%

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

      4. +-commutative 93.9%

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

   10. Applied egg-rr 93.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.1%

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

   2. associate-/l/ 90.9%

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

   3. +-commutative 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-+l+ 90.9%

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

   7. associate-+l+ 90.9%

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

   8. fma-def 90.9%

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

3. Simplified 90.9%

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

   1. fma-def 90.9%

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

   2. associate-+r+ 90.9%

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

   3. associate-+r+ 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. *-commutative 90.9%

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

   7. associate-+l+ 90.9%

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

   8. distribute-rgt1-in 90.9%

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

   9. associate-+r+ 90.9%

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

   10. +-commutative 90.9%

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

   11. *-lft-identity 90.9%

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

   12. *-rgt-identity 90.9%

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

   13. *-lft-identity 90.9%

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

   14. distribute-lft-in 90.9%

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

   15. +-commutative 90.9%

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

5. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-+r+ 99.1%

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

   3. +-commutative 99.1%

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

   4. associate-*r/ 97.3%

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

   5. *-commutative 97.3%

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

   6. associate-*l/ 97.3%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{t_1 \cdot \left(t_1 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_1}}{\frac{t_0}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 4.3e+17)
     (/ (+ alpha (+ 1.0 beta)) (* t_1 (* t_1 t_0)))
     (/ (/ 1.0 t_1) (/ t_0 (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.3e+17) {
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_1 * t_0));
	} else {
		tmp = (1.0 / t_1) / (t_0 / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 4.3d+17) then
        tmp = (alpha + (1.0d0 + beta)) / (t_1 * (t_1 * t_0))
    else
        tmp = (1.0d0 / t_1) / (t_0 / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.3e+17) {
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_1 * t_0));
	} else {
		tmp = (1.0 / t_1) / (t_0 / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 4.3e+17:
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_1 * t_0))
	else:
		tmp = (1.0 / t_1) / (t_0 / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 4.3e+17)
		tmp = Float64(Float64(alpha + Float64(1.0 + beta)) / Float64(t_1 * Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(1.0 / t_1) / Float64(t_0 / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 4.3e+17)
		tmp = (alpha + (1.0 + beta)) / (t_1 * (t_1 * t_0));
	else
		tmp = (1.0 / t_1) / (t_0 / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.3e+17], N[(N[(alpha + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] / N[(t$95$0 / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.3e17

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/l/ 95.2%

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

      3. +-commutative 95.2%

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

      4. *-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+l+ 95.2%

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

      7. associate-+l+ 95.2%

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

      8. fma-def 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in alpha around 0 95.2%

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

   if 4.3e17 < beta

   1. Initial program 91.8%

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

      1. associate-/l/ 90.5%

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

      2. associate-/l/ 80.4%

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

      3. +-commutative 80.4%

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

      4. *-commutative 80.4%

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

      5. +-commutative 80.4%

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

      6. associate-+l+ 80.4%

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

      7. associate-+l+ 80.4%

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

      8. fma-def 80.4%

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

   3. Simplified 80.4%

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

      1. fma-def 80.4%

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

      2. associate-+r+ 80.4%

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

      3. associate-+r+ 80.4%

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

      4. *-commutative 80.4%

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

      5. +-commutative 80.4%

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

      6. *-commutative 80.4%

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

      7. associate-+l+ 80.4%

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

      8. distribute-rgt1-in 80.4%

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

      9. associate-+r+ 80.4%

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

      10. +-commutative 80.4%

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

      11. *-lft-identity 80.4%

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

      12. *-rgt-identity 80.4%

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

      13. *-lft-identity 80.4%

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

      14. distribute-lft-in 80.4%

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

      15. +-commutative 80.4%

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

   5. Applied egg-rr 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. +-commutative 97.2%

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

      4. associate-*r/ 90.9%

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

      5. *-commutative 90.9%

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

      6. associate-*l/ 90.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in beta around inf 93.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 3\right)}{1 + \alpha}}\\ \end{array} \]

Alternative 4: 96.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.1%

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

   2. associate-/l/ 90.9%

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

   3. +-commutative 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-+l+ 90.9%

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

   7. associate-+l+ 90.9%

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

   8. fma-def 90.9%

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

3. Simplified 90.9%

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

   1. fma-def 90.9%

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

   2. associate-+r+ 90.9%

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

   3. associate-+r+ 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. *-commutative 90.9%

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

   7. associate-+l+ 90.9%

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

   8. distribute-rgt1-in 90.9%

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

   9. associate-+r+ 90.9%

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

   10. +-commutative 90.9%

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

   11. *-lft-identity 90.9%

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

   12. *-rgt-identity 90.9%

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

   13. *-lft-identity 90.9%

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

   14. distribute-lft-in 90.9%

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

   15. +-commutative 90.9%

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

5. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-+r+ 99.1%

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

   3. +-commutative 99.1%

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

   4. associate-*r/ 97.3%

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

   5. *-commutative 97.3%

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

   6. associate-*l/ 97.3%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 5: 96.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.0%

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

   2. associate-/l/ 90.9%

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

   3. associate-/r* 97.1%

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

3. Simplified 97.1%

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

   1. frac-2neg 97.1%

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

   2. distribute-frac-neg 97.1%

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

   3. neg-sub0 97.1%

      \[\leadsto \frac{\color{blue}{0 - \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{-\left(\left(\alpha + \beta\right) + 3\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   4. *-commutative 97.1%

      \[\leadsto \frac{0 - \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{-\left(\left(\alpha + \beta\right) + 3\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   5. +-commutative 97.1%

      \[\leadsto \frac{0 - \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{-\color{blue}{\left(3 + \left(\alpha + \beta\right)\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   6. distribute-neg-in 97.1%

      \[\leadsto \frac{0 - \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{\left(-3\right) + \left(-\left(\alpha + \beta\right)\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   7. unsub-neg 97.1%

      \[\leadsto \frac{0 - \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{\left(-3\right) - \left(\alpha + \beta\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   8. metadata-eval 97.1%

      \[\leadsto \frac{0 - \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{-3} - \left(\alpha + \beta\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

5. Applied egg-rr 97.1%

   \[\leadsto \frac{\color{blue}{0 - \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{-3 - \left(\alpha + \beta\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
6. Step-by-step derivation

   1. neg-sub0 97.1%

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

   2. associate-/l* 98.9%

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

   3. +-commutative 98.9%

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

   4. +-commutative 98.9%

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

   5. +-commutative 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 6: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{t_0}{\frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{\alpha + \left(\beta + 3\right)}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ alpha (+ beta 2.0)))))
   (if (<= beta 8.2e+40)
     (/ t_0 (/ (* (+ beta 2.0) (+ beta 3.0)) (+ 1.0 beta)))
     (/ t_0 (/ (+ alpha (+ beta 3.0)) (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 1.0 / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 8.2e+40) {
		tmp = t_0 / (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta));
	} else {
		tmp = t_0 / ((alpha + (beta + 3.0)) / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (alpha + (beta + 2.0d0))
    if (beta <= 8.2d+40) then
        tmp = t_0 / (((beta + 2.0d0) * (beta + 3.0d0)) / (1.0d0 + beta))
    else
        tmp = t_0 / ((alpha + (beta + 3.0d0)) / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 1.0 / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 8.2e+40) {
		tmp = t_0 / (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta));
	} else {
		tmp = t_0 / ((alpha + (beta + 3.0)) / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 1.0 / (alpha + (beta + 2.0))
	tmp = 0
	if beta <= 8.2e+40:
		tmp = t_0 / (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta))
	else:
		tmp = t_0 / ((alpha + (beta + 3.0)) / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(1.0 / Float64(alpha + Float64(beta + 2.0)))
	tmp = 0.0
	if (beta <= 8.2e+40)
		tmp = Float64(t_0 / Float64(Float64(Float64(beta + 2.0) * Float64(beta + 3.0)) / Float64(1.0 + beta)));
	else
		tmp = Float64(t_0 / Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 1.0 / (alpha + (beta + 2.0));
	tmp = 0.0;
	if (beta <= 8.2e+40)
		tmp = t_0 / (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta));
	else
		tmp = t_0 / ((alpha + (beta + 3.0)) / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(1.0 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8.2e+40], N[(t$95$0 / N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2000000000000003e40

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/l/ 95.3%

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

      3. +-commutative 95.3%

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

      4. *-commutative 95.3%

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

      5. +-commutative 95.3%

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

      6. associate-+l+ 95.3%

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

      7. associate-+l+ 95.3%

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

      8. fma-def 95.3%

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

   3. Simplified 95.3%

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

      1. fma-def 95.3%

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

      2. associate-+r+ 95.3%

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

      3. associate-+r+ 95.3%

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

      4. *-commutative 95.3%

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

      5. +-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. associate-+l+ 95.3%

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

      8. distribute-rgt1-in 95.3%

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

      9. associate-+r+ 95.3%

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

      10. +-commutative 95.3%

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

      11. *-lft-identity 95.3%

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

      12. *-rgt-identity 95.3%

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

      13. *-lft-identity 95.3%

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

      14. distribute-lft-in 95.3%

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

      15. +-commutative 95.3%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l/ 99.9%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in alpha around 0 68.6%

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

   if 8.2000000000000003e40 < beta

   1. Initial program 91.5%

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

      1. associate-/l/ 90.1%

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

      2. associate-/l/ 79.6%

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

      3. +-commutative 79.6%

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

      4. *-commutative 79.6%

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

      5. +-commutative 79.6%

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

      6. associate-+l+ 79.6%

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

      7. associate-+l+ 79.6%

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

      8. fma-def 79.6%

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

   3. Simplified 79.6%

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

      1. fma-def 79.6%

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

      2. associate-+r+ 79.6%

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

      3. associate-+r+ 79.6%

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

      4. *-commutative 79.6%

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

      5. +-commutative 79.6%

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

      6. *-commutative 79.6%

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

      7. associate-+l+ 79.6%

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

      8. distribute-rgt1-in 79.6%

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

      9. associate-+r+ 79.6%

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

      10. +-commutative 79.6%

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

      11. *-lft-identity 79.6%

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

      12. *-rgt-identity 79.6%

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

      13. *-lft-identity 79.6%

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

      14. distribute-lft-in 79.6%

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

      15. +-commutative 79.6%

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

   5. Applied egg-rr 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-+r+ 97.0%

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

      3. +-commutative 97.0%

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

      4. associate-*r/ 90.5%

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

      5. *-commutative 90.5%

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

      6. associate-*l/ 90.5%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in beta around inf 94.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 3\right)}{1 + \alpha}}\\ \end{array} \]

Alternative 7: 95.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.1%

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

   2. associate-/l/ 90.9%

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

   3. +-commutative 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-+l+ 90.9%

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

   7. associate-+l+ 90.9%

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

   8. fma-def 90.9%

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

3. Simplified 90.9%

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

   1. fma-def 90.9%

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

   2. associate-+r+ 90.9%

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

   3. associate-+r+ 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. *-commutative 90.9%

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

   7. associate-+l+ 90.9%

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

   8. distribute-rgt1-in 90.9%

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

   9. associate-+r+ 90.9%

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

   10. +-commutative 90.9%

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

   11. *-lft-identity 90.9%

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

   12. *-rgt-identity 90.9%

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

   13. *-lft-identity 90.9%

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

   14. distribute-lft-in 90.9%

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

   15. +-commutative 90.9%

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

5. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-+r+ 99.1%

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

   3. +-commutative 99.1%

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

   4. associate-*r/ 97.3%

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

   5. *-commutative 97.3%

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

   6. associate-*l/ 97.3%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in alpha around 0 75.2%

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

   1. +-commutative 75.2%

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

   2. +-commutative 75.2%

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

10. Simplified 75.2%

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

11. Final simplification 75.2%

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

Alternative 8: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{0.3333333333333333 + \alpha \cdot 0.2222222222222222}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.9)
     (/ (+ 0.3333333333333333 (* alpha 0.2222222222222222)) (* t_0 t_0))
     (/ (/ 1.0 t_0) (/ beta (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.9) {
		tmp = (0.3333333333333333 + (alpha * 0.2222222222222222)) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_0) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3.9d0) then
        tmp = (0.3333333333333333d0 + (alpha * 0.2222222222222222d0)) / (t_0 * t_0)
    else
        tmp = (1.0d0 / t_0) / (beta / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.9) {
		tmp = (0.3333333333333333 + (alpha * 0.2222222222222222)) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_0) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.9:
		tmp = (0.3333333333333333 + (alpha * 0.2222222222222222)) / (t_0 * t_0)
	else:
		tmp = (1.0 / t_0) / (beta / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.9)
		tmp = Float64(Float64(0.3333333333333333 + Float64(alpha * 0.2222222222222222)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) / Float64(beta / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.9)
		tmp = (0.3333333333333333 + (alpha * 0.2222222222222222)) / (t_0 * t_0);
	else
		tmp = (1.0 / t_0) / (beta / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.9], N[(N[(0.3333333333333333 + N[(alpha * 0.2222222222222222), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999991

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/l/ 95.2%

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

      3. associate-/r* 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in beta around 0 98.0%

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

   5. Taylor expanded in alpha around 0 82.2%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 + 0.2222222222222222 \cdot \alpha}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 82.2%

         \[\leadsto \frac{0.3333333333333333 + \color{blue}{\alpha \cdot 0.2222222222222222}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   7. Simplified 82.2%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 + \alpha \cdot 0.2222222222222222}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   if 3.89999999999999991 < beta

   1. Initial program 92.0%

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

      1. associate-/l/ 90.8%

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

      2. associate-/l/ 80.9%

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

      3. +-commutative 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. associate-+l+ 80.9%

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

      7. associate-+l+ 80.9%

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

      8. fma-def 80.9%

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

   3. Simplified 80.9%

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

      1. fma-def 80.9%

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

      2. associate-+r+ 80.9%

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

      3. associate-+r+ 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. *-commutative 80.9%

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

      7. associate-+l+ 80.9%

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

      8. distribute-rgt1-in 80.9%

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

      9. associate-+r+ 80.9%

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

      10. +-commutative 80.9%

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

      11. *-lft-identity 80.9%

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

      12. *-rgt-identity 80.9%

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

      13. *-lft-identity 80.9%

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

      14. distribute-lft-in 80.9%

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

      15. +-commutative 80.9%

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

   5. Applied egg-rr 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. +-commutative 97.2%

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

      4. associate-*r/ 91.2%

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

      5. *-commutative 91.2%

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

      6. associate-*l/ 91.1%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in beta around inf 91.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{0.3333333333333333 + \alpha \cdot 0.2222222222222222}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 9: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10.5:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10.5)
   (/ (+ 0.5 (* alpha 0.25)) (* (+ 3.0 (+ alpha beta)) (+ 2.0 (+ alpha beta))))
   (/ (/ 1.0 (+ alpha (+ beta 2.0))) (/ beta (+ 1.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.5) {
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 10.5d0) then
        tmp = (0.5d0 + (alpha * 0.25d0)) / ((3.0d0 + (alpha + beta)) * (2.0d0 + (alpha + beta)))
    else
        tmp = (1.0d0 / (alpha + (beta + 2.0d0))) / (beta / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.5) {
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 10.5:
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)))
	else:
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10.5)
		tmp = Float64(Float64(0.5 + Float64(alpha * 0.25)) / Float64(Float64(3.0 + Float64(alpha + beta)) * Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 2.0))) / Float64(beta / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 10.5)
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	else
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 10.5], N[(N[(0.5 + N[(alpha * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 10.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.0%

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

   5. Taylor expanded in alpha around 0 82.2%

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

      1. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   if 10.5 < beta

   1. Initial program 92.0%

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

      1. associate-/l/ 90.8%

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

      2. associate-/l/ 80.9%

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

      3. +-commutative 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. associate-+l+ 80.9%

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

      7. associate-+l+ 80.9%

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

      8. fma-def 80.9%

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

   3. Simplified 80.9%

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

      1. fma-def 80.9%

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

      2. associate-+r+ 80.9%

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

      3. associate-+r+ 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. *-commutative 80.9%

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

      7. associate-+l+ 80.9%

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

      8. distribute-rgt1-in 80.9%

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

      9. associate-+r+ 80.9%

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

      10. +-commutative 80.9%

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

      11. *-lft-identity 80.9%

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

      12. *-rgt-identity 80.9%

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

      13. *-lft-identity 80.9%

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

      14. distribute-lft-in 80.9%

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

      15. +-commutative 80.9%

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

   5. Applied egg-rr 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. +-commutative 97.2%

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

      4. associate-*r/ 91.2%

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

      5. *-commutative 91.2%

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

      6. associate-*l/ 91.1%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in beta around inf 91.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10.5:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 10: 97.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 3\right)}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (/ (+ 0.5 (* alpha 0.25)) (* (+ 3.0 (+ alpha beta)) (+ 2.0 (+ alpha beta))))
   (/
    (/ 1.0 (+ alpha (+ beta 2.0)))
    (/ (+ alpha (+ beta 3.0)) (+ 1.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = (1.0 / (alpha + (beta + 2.0))) / ((alpha + (beta + 3.0)) / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d0) then
        tmp = (0.5d0 + (alpha * 0.25d0)) / ((3.0d0 + (alpha + beta)) * (2.0d0 + (alpha + beta)))
    else
        tmp = (1.0d0 / (alpha + (beta + 2.0d0))) / ((alpha + (beta + 3.0d0)) / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = (1.0 / (alpha + (beta + 2.0))) / ((alpha + (beta + 3.0)) / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)))
	else:
		tmp = (1.0 / (alpha + (beta + 2.0))) / ((alpha + (beta + 3.0)) / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(Float64(0.5 + Float64(alpha * 0.25)) / Float64(Float64(3.0 + Float64(alpha + beta)) * Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = (0.5 + (alpha * 0.25)) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	else
		tmp = (1.0 / (alpha + (beta + 2.0))) / ((alpha + (beta + 3.0)) / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(N[(0.5 + N[(alpha * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.5%

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

   5. Taylor expanded in alpha around 0 82.6%

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

      1. *-commutative 82.6%

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

   7. Simplified 82.6%

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

   if 1.69999999999999996 < beta

   1. Initial program 92.1%

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

      1. associate-/l/ 90.9%

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

      2. associate-/l/ 81.2%

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

      3. +-commutative 81.2%

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

      4. *-commutative 81.2%

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

      5. +-commutative 81.2%

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

      6. associate-+l+ 81.2%

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

      7. associate-+l+ 81.2%

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

      8. fma-def 81.2%

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

   3. Simplified 81.2%

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

      1. fma-def 81.2%

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

      2. associate-+r+ 81.2%

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

      3. associate-+r+ 81.2%

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

      4. *-commutative 81.2%

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

      5. +-commutative 81.2%

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

      6. *-commutative 81.2%

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

      7. associate-+l+ 81.2%

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

      8. distribute-rgt1-in 81.2%

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

      9. associate-+r+ 81.2%

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

      10. +-commutative 81.2%

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

      11. *-lft-identity 81.2%

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

      12. *-rgt-identity 81.2%

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

      13. *-lft-identity 81.2%

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

      14. distribute-lft-in 81.2%

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

      15. +-commutative 81.2%

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

   5. Applied egg-rr 97.3%

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

      1. *-commutative 97.3%

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

      2. associate-+r+ 97.3%

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

      3. +-commutative 97.3%

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

      4. associate-*r/ 91.3%

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

      5. *-commutative 91.3%

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

      6. associate-*l/ 91.2%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in beta around inf 90.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 3\right)}{1 + \alpha}}\\ \end{array} \]

Alternative 11: 96.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ 1.0 (+ alpha (+ beta 2.0))) (/ beta (+ 1.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = (1.0d0 / (alpha + (beta + 2.0d0))) / (beta / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 2.0))) / Float64(beta / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = (1.0 / (alpha + (beta + 2.0))) / (beta / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.0%

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

   5. Taylor expanded in alpha around 0 66.1%

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

      1. associate-/r* 66.1%

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

      2. +-commutative 66.1%

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

      3. +-commutative 66.1%

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

   7. Simplified 66.1%

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

   if 6 < beta

   1. Initial program 92.0%

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

      1. associate-/l/ 90.8%

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

      2. associate-/l/ 80.9%

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

      3. +-commutative 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. associate-+l+ 80.9%

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

      7. associate-+l+ 80.9%

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

      8. fma-def 80.9%

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

   3. Simplified 80.9%

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

      1. fma-def 80.9%

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

      2. associate-+r+ 80.9%

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

      3. associate-+r+ 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. *-commutative 80.9%

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

      7. associate-+l+ 80.9%

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

      8. distribute-rgt1-in 80.9%

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

      9. associate-+r+ 80.9%

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

      10. +-commutative 80.9%

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

      11. *-lft-identity 80.9%

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

      12. *-rgt-identity 80.9%

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

      13. *-lft-identity 80.9%

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

      14. distribute-lft-in 80.9%

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

      15. +-commutative 80.9%

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

   5. Applied egg-rr 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. +-commutative 97.2%

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

      4. associate-*r/ 91.2%

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

      5. *-commutative 91.2%

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

      6. associate-*l/ 91.1%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in beta around inf 91.3%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 12: 96.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/ 0.5 (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.0d0) then
        tmp = 0.5d0 / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.0:
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = Float64(0.5 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.0)
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.0], N[(0.5 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.0%

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

   5. Taylor expanded in alpha around 0 66.1%

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

   if 9 < beta

   1. Initial program 92.0%

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

      1. associate-/l/ 90.8%

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

      2. associate-/l/ 80.9%

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

      3. +-commutative 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. associate-+l+ 80.9%

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

      7. associate-+l+ 80.9%

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

      8. fma-def 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in beta around inf 91.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 91.9%

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

   6. Simplified 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-/r* 91.2%

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

      3. div-inv 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. un-div-inv 91.2%

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

      2. +-commutative 91.2%

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

   10. Applied egg-rr 91.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13: 96.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.2)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.2d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.2:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.2)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.2)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.2], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.1999999999999993

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.0%

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

   5. Taylor expanded in alpha around 0 66.1%

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

      1. associate-/r* 66.1%

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

      2. +-commutative 66.1%

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

      3. +-commutative 66.1%

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

   7. Simplified 66.1%

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

   if 9.1999999999999993 < beta

   1. Initial program 92.0%

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

      1. associate-/l/ 90.8%

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

      2. associate-/l/ 80.9%

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

      3. +-commutative 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. associate-+l+ 80.9%

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

      7. associate-+l+ 80.9%

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

      8. fma-def 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in beta around inf 91.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 91.9%

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

   6. Simplified 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-/r* 91.2%

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

      3. div-inv 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. un-div-inv 91.2%

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

      2. +-commutative 91.2%

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

   10. Applied egg-rr 91.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14: 96.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10.0)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ 1.0 beta) (/ beta (+ 1.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 10.0d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = (1.0d0 / beta) / (beta / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 10.0:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = (1.0 / beta) / (beta / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10.0)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 10.0)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = (1.0 / beta) / (beta / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 10.0], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 10

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.0%

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

   5. Taylor expanded in alpha around 0 66.1%

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

      1. associate-/r* 66.1%

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

      2. +-commutative 66.1%

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

      3. +-commutative 66.1%

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

   7. Simplified 66.1%

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

   if 10 < beta

   1. Initial program 92.0%

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

      1. associate-/l/ 90.8%

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

      2. associate-/l/ 80.9%

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

      3. +-commutative 80.9%

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

      4. *-commutative 80.9%

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

      5. +-commutative 80.9%

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

      6. associate-+l+ 80.9%

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

      7. associate-+l+ 80.9%

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

      8. fma-def 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in beta around inf 91.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 91.9%

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

   6. Simplified 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-/r* 91.2%

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

      3. div-inv 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. clear-num 91.1%

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

      2. associate-*l/ 91.2%

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

      3. div-inv 91.2%

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

      4. +-commutative 91.2%

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

   10. Applied egg-rr 91.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 15: 54.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.1e-37) (/ 1.0 (* beta beta)) (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.1e-37) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.1d-37) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.1e-37) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 2.1e-37:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.1e-37)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.1e-37)
		tmp = 1.0 / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 2.1e-37], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.1000000000000001e-37

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in beta around 0 87.0%

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

   5. Taylor expanded in beta around inf 23.5%

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

      1. *-commutative 23.5%

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

      2. unpow2 23.5%

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

   7. Simplified 23.5%

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

   8. Taylor expanded in alpha around inf 34.2%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 34.2%

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

   10. Simplified 34.2%

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

   if 2.1000000000000001e-37 < alpha

   1. Initial program 92.9%

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

      1. associate-/l/ 91.7%

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

      2. associate-/l/ 80.2%

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

      3. +-commutative 80.2%

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

      4. *-commutative 80.2%

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

      5. +-commutative 80.2%

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

      6. associate-+l+ 80.2%

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

      7. associate-+l+ 80.2%

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

      8. fma-def 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in beta around inf 21.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 21.7%

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

   6. Simplified 21.7%

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

   7. Taylor expanded in alpha around inf 21.7%

      \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 21.7%

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

   9. Simplified 21.7%

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

   10. Taylor expanded in alpha around 0 21.7%

       \[\leadsto \color{blue}{\frac{\alpha}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 21.7%

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

       2. associate-/r* 21.1%

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

   12. Simplified 21.1%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 16: 53.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.1%

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

   2. associate-/l/ 90.9%

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

   3. +-commutative 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-+l+ 90.9%

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

   7. associate-+l+ 90.9%

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

   8. fma-def 90.9%

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

3. Simplified 90.9%

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

4. Taylor expanded in beta around inf 29.9%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 29.9%

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

6. Simplified 29.9%

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

7. Final simplification 29.9%

   \[\leadsto \frac{1 + \alpha}{\beta \cdot \beta} \]

Alternative 17: 56.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.1%

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

   2. associate-/l/ 90.9%

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

   3. +-commutative 90.9%

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

   4. *-commutative 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-+l+ 90.9%

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

   7. associate-+l+ 90.9%

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

   8. fma-def 90.9%

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

3. Simplified 90.9%

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

4. Taylor expanded in beta around inf 29.9%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 29.9%

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

6. Simplified 29.9%

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

   1. +-commutative 29.9%

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

   2. associate-/r* 29.7%

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

   3. div-inv 29.6%

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

8. Applied egg-rr 29.6%

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

   1. un-div-inv 29.7%

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

   2. +-commutative 29.7%

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

10. Applied egg-rr 29.7%

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

11. Final simplification 29.7%

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

Alternative 18: 34.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.5 (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / (beta * 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 / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / (beta * beta);
end

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.1%

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

   2. +-commutative 97.1%

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

   3. *-commutative 97.1%

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

   4. +-commutative 97.1%

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

   5. +-commutative 97.1%

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

   6. +-commutative 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in beta around 0 89.3%

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

5. Taylor expanded in beta around inf 22.3%

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

   1. *-commutative 22.3%

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

   2. unpow2 22.3%

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

7. Simplified 22.3%

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

8. Taylor expanded in alpha around 0 21.9%

   \[\leadsto \color{blue}{\frac{0.5}{{\beta}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 21.9%

      \[\leadsto \frac{0.5}{\color{blue}{\beta \cdot \beta}} \]

10. Simplified 21.9%

    \[\leadsto \color{blue}{\frac{0.5}{\beta \cdot \beta}} \]

11. Final simplification 21.9%

    \[\leadsto \frac{0.5}{\beta \cdot \beta} \]

Alternative 19: 50.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.1%

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

   2. +-commutative 97.1%

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

   3. *-commutative 97.1%

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

   4. +-commutative 97.1%

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

   5. +-commutative 97.1%

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

   6. +-commutative 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in beta around 0 89.3%

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

5. Taylor expanded in beta around inf 22.3%

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

   1. *-commutative 22.3%

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

   2. unpow2 22.3%

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

7. Simplified 22.3%

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

8. Taylor expanded in alpha around inf 28.9%

   \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 28.9%

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

10. Simplified 28.9%

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

11. Final simplification 28.9%

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

Alternative 20: 2.5% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.2222222222222222}{\alpha} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.2222222222222222 alpha))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.2222222222222222 / alpha;
}

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 = 0.2222222222222222d0 / alpha
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.2222222222222222 / alpha;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.2222222222222222 / alpha

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.2222222222222222 / alpha)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.2222222222222222 / alpha;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.2222222222222222 / alpha), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

   1. associate-/l/ 97.0%

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

   2. associate-/l/ 90.9%

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

   3. associate-/r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in beta around 0 89.2%

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

5. Taylor expanded in alpha around 0 78.2%

   \[\leadsto \frac{\color{blue}{0.3333333333333333 + 0.2222222222222222 \cdot \alpha}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 78.2%

      \[\leadsto \frac{0.3333333333333333 + \color{blue}{\alpha \cdot 0.2222222222222222}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

7. Simplified 78.2%

   \[\leadsto \frac{\color{blue}{0.3333333333333333 + \alpha \cdot 0.2222222222222222}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

8. Taylor expanded in alpha around inf 3.9%

   \[\leadsto \color{blue}{\frac{0.2222222222222222}{\alpha}} \]

9. Final simplification 3.9%

   \[\leadsto \frac{0.2222222222222222}{\alpha} \]

Reproduce

herbie shell --seed 2023297 
(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)))