Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.7%

Time: 15.0s

Alternatives: 13

Speedup: 2.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_1}}{t_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

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 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.5e+149)
     (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_1) (* t_1 t_0)))
     (* (/ (+ 1.0 alpha) (+ beta (+ 3.0 (* alpha 2.0)))) (/ 1.0 t_0)))))

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 <= 2.5e+149) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 2.5d+149) then
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / t_1) / (t_1 * t_0))
    else
        tmp = ((1.0d0 + alpha) / (beta + (3.0d0 + (alpha * 2.0d0)))) * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.5e+149) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / t_0);
	}
	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 <= 2.5e+149:
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_1 * t_0))
	else:
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / t_0)
	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 <= 2.5e+149)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / t_1) / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(3.0 + Float64(alpha * 2.0)))) * Float64(1.0 / t_0));
	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 <= 2.5e+149)
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	else
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5e+149], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.49999999999999995e149

   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/ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. +-commutative 98.2%

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

      4. associate-+r+ 98.2%

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

      5. associate-+l+ 98.2%

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

      6. distribute-rgt1-in 98.2%

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

      7. *-rgt-identity 98.2%

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

      8. distribute-lft-out 98.2%

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

      9. +-commutative 98.2%

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

      10. associate-*l/ 98.7%

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

      11. *-commutative 98.7%

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

      12. associate-*r/ 96.0%

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

   3. Simplified 96.0%

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

   if 2.49999999999999995e149 < beta

   1. Initial program 79.5%

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

      1. div-inv 79.5%

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

      2. +-commutative 79.5%

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

      3. associate-+l+ 79.5%

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

      4. *-commutative 79.5%

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

      5. metadata-eval 79.5%

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

      6. associate-+r+ 79.5%

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

      7. metadata-eval 79.5%

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

      8. associate-+r+ 79.5%

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

   3. Applied egg-rr 79.5%

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

      1. associate-*l/ 79.5%

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

      2. associate-*r/ 79.5%

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

      3. *-rgt-identity 79.5%

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

      4. associate-+r+ 79.5%

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

      5. *-rgt-identity 79.5%

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

      6. +-commutative 79.5%

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

      7. distribute-rgt1-in 79.5%

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

      8. distribute-lft-in 79.5%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+r+ 99.8%

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

      13. +-commutative 99.8%

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

      14. associate-+r+ 99.8%

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

   5. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. associate-+r+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+r+ 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in beta around inf 94.2%

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

       1. *-commutative 94.2%

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

   12. Simplified 94.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 2: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.6%

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

   1. div-inv 95.5%

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

   2. +-commutative 95.5%

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

   3. associate-+l+ 95.5%

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

   4. *-commutative 95.5%

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

   5. metadata-eval 95.5%

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

   6. associate-+r+ 95.5%

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

   7. metadata-eval 95.5%

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

   8. associate-+r+ 95.5%

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

3. Applied egg-rr 95.5%

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

   1. associate-*l/ 95.5%

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

   2. associate-*r/ 95.6%

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

   3. *-rgt-identity 95.6%

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

   4. associate-+r+ 95.6%

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

   5. *-rgt-identity 95.6%

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

   6. +-commutative 95.6%

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

   7. distribute-rgt1-in 95.6%

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

   8. distribute-lft-in 95.6%

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

   9. associate-*r/ 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

   12. associate-+r+ 99.7%

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

   13. +-commutative 99.7%

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

   14. associate-+r+ 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 93.6%

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

   2. associate-+l+ 93.6%

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

   3. +-commutative 93.6%

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

   4. associate-+r+ 93.6%

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

   5. associate-+l+ 93.6%

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

   6. distribute-rgt1-in 93.6%

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

   7. *-rgt-identity 93.6%

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

   8. distribute-lft-out 93.6%

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

   9. +-commutative 93.6%

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

   10. associate-*l/ 95.9%

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

   11. *-commutative 95.9%

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

   12. associate-*r/ 93.7%

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

3. Simplified 93.7%

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

   1. associate-*r/ 95.9%

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

   2. +-commutative 95.9%

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

5. Applied egg-rr 95.9%

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

   1. +-commutative 95.9%

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

   2. *-commutative 95.9%

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

   3. +-commutative 95.9%

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

   4. associate-*r/ 95.9%

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

   5. associate-/r* 99.7%

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

   6. +-commutative 99.7%

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

   7. associate-+r+ 99.7%

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

   8. +-commutative 99.7%

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

   9. +-commutative 99.7%

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

   10. associate-+r+ 99.7%

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

   11. +-commutative 99.7%

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

   12. associate-+r+ 99.7%

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

   13. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 93.6%

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

   2. associate-+l+ 93.6%

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

   3. +-commutative 93.6%

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

   4. associate-+r+ 93.6%

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

   5. associate-+l+ 93.6%

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

   6. distribute-rgt1-in 93.6%

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

   7. *-rgt-identity 93.6%

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

   8. distribute-lft-out 93.6%

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

   9. +-commutative 93.6%

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

   10. associate-*l/ 95.9%

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

   11. *-commutative 95.9%

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

   12. associate-*r/ 93.7%

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

3. Simplified 93.7%

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

   1. associate-*r/ 95.9%

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

   2. +-commutative 95.9%

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

5. Applied egg-rr 95.9%

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

   1. +-commutative 95.9%

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

   2. *-commutative 95.9%

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

   3. +-commutative 95.9%

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

   4. associate-*r/ 95.9%

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

   5. associate-/r* 99.7%

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

   6. +-commutative 99.7%

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

   7. associate-+r+ 99.7%

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

   8. +-commutative 99.7%

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

   9. +-commutative 99.7%

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

   10. associate-+r+ 99.7%

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

   11. +-commutative 99.7%

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

   12. associate-+r+ 99.7%

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

   13. +-commutative 99.7%

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

7. Simplified 99.7%

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

   1. associate-*r/ 99.8%

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

   2. associate-+r+ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 29000000:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 29000000.0)
   (* (+ 1.0 beta) (/ (/ 1.0 (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))
   (*
    (/ (+ 1.0 alpha) (+ beta (+ 3.0 (* alpha 2.0))))
    (/ 1.0 (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 29000000.0) {
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 29000000.0d0) then
        tmp = (1.0d0 + beta) * ((1.0d0 / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + (3.0d0 + (alpha * 2.0d0)))) * (1.0d0 / (alpha + (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 29000000.0) {
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 29000000.0:
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 29000000.0)
		tmp = Float64(Float64(1.0 + beta) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(3.0 + Float64(alpha * 2.0)))) * Float64(1.0 / Float64(alpha + Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 29000000.0)
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 29000000.0], N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9e7

   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/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in alpha around 0 86.4%

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

   5. Taylor expanded in alpha around 0 67.5%

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

   if 2.9e7 < beta

   1. Initial program 88.8%

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

      1. div-inv 88.7%

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

      2. +-commutative 88.7%

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

      3. associate-+l+ 88.7%

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

      4. *-commutative 88.7%

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

      5. metadata-eval 88.7%

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

      6. associate-+r+ 88.7%

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

      7. metadata-eval 88.7%

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

      8. associate-+r+ 88.7%

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

   3. Applied egg-rr 88.7%

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

      1. associate-*l/ 88.7%

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

      2. associate-*r/ 88.8%

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

      3. *-rgt-identity 88.8%

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

      4. associate-+r+ 88.8%

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

      5. *-rgt-identity 88.8%

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

      6. +-commutative 88.8%

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

      7. distribute-rgt1-in 88.8%

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

      8. distribute-lft-in 88.8%

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

      9. associate-*r/ 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. +-commutative 99.6%

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

      14. associate-+r+ 99.6%

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

   5. Simplified 99.6%

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

      1. div-inv 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+r+ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+l+ 99.5%

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

      7. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in beta around inf 83.9%

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

       1. *-commutative 83.9%

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

   12. Simplified 83.9%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 29000000:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10500000:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10500000.0)
   (*
    (/ (+ 1.0 beta) (+ beta (+ alpha 2.0)))
    (/ 1.0 (* (+ beta 2.0) (+ beta 3.0))))
   (*
    (/ (+ 1.0 alpha) (+ beta (+ 3.0 (* alpha 2.0))))
    (/ 1.0 (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10500000.0) {
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * (1.0 / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 10500000.0d0) then
        tmp = ((1.0d0 + beta) / (beta + (alpha + 2.0d0))) * (1.0d0 / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + (3.0d0 + (alpha * 2.0d0)))) * (1.0d0 / (alpha + (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10500000.0) {
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * (1.0 / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 10500000.0:
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * (1.0 / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10500000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + Float64(alpha + 2.0))) * Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(3.0 + Float64(alpha * 2.0)))) * Float64(1.0 / Float64(alpha + Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 10500000.0)
		tmp = ((1.0 + beta) / (beta + (alpha + 2.0))) * (1.0 / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (beta + (3.0 + (alpha * 2.0)))) * (1.0 / (alpha + (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 10500000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.05e7

   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/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*l/ 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-*r/ 97.0%

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

   3. Simplified 97.0%

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

      1. associate-*r/ 99.6%

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

      2. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*r/ 99.5%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+r+ 99.8%

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

      13. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in alpha around 0 68.7%

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

   if 1.05e7 < beta

   1. Initial program 88.8%

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

      1. div-inv 88.7%

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

      2. +-commutative 88.7%

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

      3. associate-+l+ 88.7%

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

      4. *-commutative 88.7%

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

      5. metadata-eval 88.7%

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

      6. associate-+r+ 88.7%

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

      7. metadata-eval 88.7%

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

      8. associate-+r+ 88.7%

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

   3. Applied egg-rr 88.7%

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

      1. associate-*l/ 88.7%

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

      2. associate-*r/ 88.8%

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

      3. *-rgt-identity 88.8%

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

      4. associate-+r+ 88.8%

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

      5. *-rgt-identity 88.8%

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

      6. +-commutative 88.8%

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

      7. distribute-rgt1-in 88.8%

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

      8. distribute-lft-in 88.8%

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

      9. associate-*r/ 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. +-commutative 99.6%

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

      14. associate-+r+ 99.6%

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

   5. Simplified 99.6%

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

      1. div-inv 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+r+ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+l+ 99.5%

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

      7. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in beta around inf 83.9%

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

       1. *-commutative 83.9%

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

   12. Simplified 83.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10500000:\\ \;\;\;\;\frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3e+15)
   (* (+ 1.0 beta) (/ (/ 1.0 (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) (+ beta (+ alpha 2.0))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+15) {
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.3d+15) then
        tmp = (1.0d0 + beta) * ((1.0d0 / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 2.0d0))) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+15) {
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.3e+15:
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3e+15)
		tmp = Float64(Float64(1.0 + beta) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 2.0))) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3e+15)
		tmp = (1.0 + beta) * ((1.0 / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.3e+15], N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3e15

   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/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.5%

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

      11. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in alpha around 0 86.6%

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

   5. Taylor expanded in alpha around 0 68.1%

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

   if 1.3e15 < beta

   1. Initial program 88.4%

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

      1. associate-/l/ 83.7%

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

      2. associate-+l+ 83.7%

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

      3. +-commutative 83.7%

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

      4. associate-+r+ 83.7%

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

      5. associate-+l+ 83.7%

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

      6. distribute-rgt1-in 83.7%

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

      7. *-rgt-identity 83.7%

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

      8. distribute-lft-out 83.7%

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

      9. +-commutative 83.7%

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

      10. associate-*l/ 89.7%

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

      11. *-commutative 89.7%

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

      12. associate-*r/ 88.2%

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

   3. Simplified 88.2%

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

      1. associate-*r/ 89.7%

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

      2. +-commutative 89.7%

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

   5. Applied egg-rr 89.7%

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

      1. +-commutative 89.7%

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

      2. *-commutative 89.7%

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

      3. +-commutative 89.7%

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

      4. associate-*r/ 89.7%

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

      5. associate-/r* 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-+r+ 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 83.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 8: 96.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 0.9:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 0.9)
   0.08333333333333333
   (/ (/ (+ 1.0 alpha) (+ beta (+ alpha 2.0))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.9) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 0.9d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 2.0d0))) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.9) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 0.9:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 0.9)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 2.0))) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 0.9)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 0.9], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 0.900000000000000022

   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/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 86.3%

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

   5. Taylor expanded in beta around 0 83.7%

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

   6. Taylor expanded in alpha around 0 64.6%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 0.900000000000000022 < beta

   1. Initial program 89.0%

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

      1. associate-/l/ 84.4%

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

      2. associate-+l+ 84.4%

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

      3. +-commutative 84.4%

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

      4. associate-+r+ 84.4%

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

      5. associate-+l+ 84.4%

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

      6. distribute-rgt1-in 84.4%

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

      7. *-rgt-identity 84.4%

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

      8. distribute-lft-out 84.4%

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

      9. +-commutative 84.4%

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

      10. associate-*l/ 90.2%

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

      11. *-commutative 90.2%

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

      12. associate-*r/ 88.7%

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

   3. Simplified 88.7%

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

      1. associate-*r/ 90.2%

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

      2. +-commutative 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. +-commutative 90.2%

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

      2. *-commutative 90.2%

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

      3. +-commutative 90.2%

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

      4. associate-*r/ 90.2%

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

      5. associate-/r* 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-+r+ 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around inf 82.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 0.9:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 9: 96.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   0.08333333333333333
   (* (/ 1.0 (+ alpha (+ beta 3.0))) (/ (+ 1.0 alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = (1.0d0 / (alpha + (beta + 3.0d0))) * ((1.0d0 + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333
	else:
		tmp = (1.0 / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = (1.0 / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], 0.08333333333333333, N[(N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   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/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 86.3%

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

   5. Taylor expanded in beta around 0 83.7%

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

   6. Taylor expanded in alpha around 0 64.6%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.2999999999999998 < beta

   1. Initial program 89.0%

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

      1. div-inv 88.9%

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

      2. +-commutative 88.9%

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

      3. associate-+l+ 88.9%

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

      4. *-commutative 88.9%

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

      5. metadata-eval 88.9%

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

      6. associate-+r+ 88.9%

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

      7. metadata-eval 88.9%

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

      8. associate-+r+ 88.9%

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

   3. Applied egg-rr 88.9%

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

      1. associate-*l/ 88.9%

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

      2. associate-*r/ 89.0%

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

      3. *-rgt-identity 89.0%

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

      4. associate-+r+ 89.0%

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

      5. *-rgt-identity 89.0%

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

      6. +-commutative 89.0%

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

      7. distribute-rgt1-in 89.0%

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

      8. distribute-lft-in 89.0%

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

      9. associate-*r/ 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. +-commutative 99.6%

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

      14. associate-+r+ 99.6%

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

   5. Simplified 99.6%

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

      1. div-inv 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-+l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-+r+ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+l+ 99.5%

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

      7. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in beta around inf 81.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \]

Alternative 10: 96.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   0.08333333333333333
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   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/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 86.3%

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

   5. Taylor expanded in beta around 0 83.7%

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

   6. Taylor expanded in alpha around 0 64.6%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.2999999999999998 < beta

   1. Initial program 89.0%

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

      1. div-inv 88.9%

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

      2. +-commutative 88.9%

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

      3. associate-+l+ 88.9%

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

      4. *-commutative 88.9%

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

      5. metadata-eval 88.9%

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

      6. associate-+r+ 88.9%

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

      7. metadata-eval 88.9%

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

      8. associate-+r+ 88.9%

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

   3. Applied egg-rr 88.9%

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

      1. associate-*l/ 88.9%

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

      2. associate-*r/ 89.0%

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

      3. *-rgt-identity 89.0%

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

      4. associate-+r+ 89.0%

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

      5. *-rgt-identity 89.0%

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

      6. +-commutative 89.0%

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

      7. distribute-rgt1-in 89.0%

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

      8. distribute-lft-in 89.0%

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

      9. associate-*r/ 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. +-commutative 99.6%

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

      14. associate-+r+ 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in beta around inf 81.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 11: 93.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6) 0.08333333333333333 (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.08333333333333333;
	} 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 <= 3.6d0) then
        tmp = 0.08333333333333333d0
    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 <= 3.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = 0.08333333333333333
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = 0.08333333333333333;
	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, 3.6], 0.08333333333333333, N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   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/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 86.3%

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

   5. Taylor expanded in beta around 0 83.3%

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

   6. Taylor expanded in alpha around 0 64.3%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 3.60000000000000009 < beta

   1. Initial program 88.9%

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

      1. div-inv 88.8%

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

      2. +-commutative 88.8%

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

      3. associate-+l+ 88.8%

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

      4. *-commutative 88.8%

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

      5. metadata-eval 88.8%

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

      6. associate-+r+ 88.8%

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

      7. metadata-eval 88.8%

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

      8. associate-+r+ 88.8%

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

   3. Applied egg-rr 88.8%

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

      1. associate-*l/ 88.9%

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

      2. associate-*r/ 88.9%

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

      3. *-rgt-identity 88.9%

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

      4. associate-+r+ 88.9%

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

      5. *-rgt-identity 88.9%

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

      6. +-commutative 88.9%

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

      7. distribute-rgt1-in 88.9%

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

      8. distribute-lft-in 88.9%

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

      9. associate-*r/ 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. +-commutative 99.6%

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

      14. associate-+r+ 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in beta around inf 77.2%

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

      1. unpow2 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12: 90.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5) 0.08333333333333333 (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.5d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.5:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.5)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.5], 0.08333333333333333, N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-+l+ 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. associate-*r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 86.3%

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

   5. Taylor expanded in beta around 0 83.3%

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

   6. Taylor expanded in alpha around 0 64.3%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 3.5 < beta

   1. Initial program 88.9%

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

      1. associate-/l/ 84.3%

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

      2. associate-+l+ 84.3%

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

      3. +-commutative 84.3%

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

      4. associate-+r+ 84.3%

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

      5. associate-+l+ 84.3%

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

      6. distribute-rgt1-in 84.3%

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

      7. *-rgt-identity 84.3%

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

      8. distribute-lft-out 84.3%

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

      9. +-commutative 84.3%

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

      10. associate-*r/ 90.0%

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

      11. associate-*r/ 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in alpha around 0 65.9%

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

   5. Taylor expanded in beta around inf 73.7%

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

      1. unpow2 73.7%

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

   7. Simplified 73.7%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 13: 44.6% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

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 = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 93.6%

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

   2. associate-+l+ 93.6%

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

   3. +-commutative 93.6%

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

   4. associate-+r+ 93.6%

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

   5. associate-+l+ 93.6%

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

   6. distribute-rgt1-in 93.6%

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

   7. *-rgt-identity 93.6%

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

   8. distribute-lft-out 93.6%

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

   9. +-commutative 93.6%

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

   10. associate-*r/ 95.8%

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

   11. associate-*r/ 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in alpha around 0 78.3%

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

5. Taylor expanded in beta around 0 54.7%

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

6. Taylor expanded in alpha around 0 40.8%

   \[\leadsto \color{blue}{0.08333333333333333} \]

7. Final simplification 40.8%

   \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023228 
(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)))