Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 18.1s

Alternatives: 19

Speedup: 3.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. associate-/l/ 92.7%

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

   2. associate-/r* 84.1%

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

   3. +-commutative 84.1%

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

   4. associate-+l+ 84.1%

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

   5. associate-+r+ 84.1%

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

   6. *-commutative 84.1%

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

   7. distribute-rgt1-in 84.1%

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

   8. +-commutative 84.1%

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

   9. *-commutative 84.1%

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

   10. distribute-rgt1-in 84.1%

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

   11. +-commutative 84.1%

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

   12. times-frac 96.0%

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

3. Simplified 96.0%

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

   1. associate-*r/ 96.1%

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

   2. +-commutative 96.1%

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

5. Applied egg-rr 96.1%

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

   1. associate-*r/ 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. clear-num 99.8%

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

   5. frac-times 99.4%

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

   6. *-un-lft-identity 99.4%

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

   7. +-commutative 99.4%

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

   8. +-commutative 99.4%

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

   9. +-commutative 99.4%

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

   10. associate-+r+ 99.4%

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

   11. +-commutative 99.4%

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

   12. +-commutative 99.4%

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

   13. +-commutative 99.4%

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

   14. +-commutative 99.4%

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

   15. associate-+r+ 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. associate-/l/ 92.7%

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

   2. associate-/r* 84.1%

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

   3. +-commutative 84.1%

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

   4. associate-+l+ 84.1%

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

   5. associate-+r+ 84.1%

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

   6. *-commutative 84.1%

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

   7. distribute-rgt1-in 84.1%

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

   8. +-commutative 84.1%

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

   9. *-commutative 84.1%

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

   10. distribute-rgt1-in 84.1%

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

   11. +-commutative 84.1%

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

   12. times-frac 96.0%

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

3. Simplified 96.0%

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

   1. associate-*r/ 96.1%

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

   2. +-commutative 96.1%

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

5. Applied egg-rr 96.1%

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

   1. associate-*r/ 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + 2}}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (*
    (/ 1.0 (+ alpha (+ beta 3.0)))
    (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (+ alpha 2.0)))
   (/
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ (+ beta 4.0) (* alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = (1.0 / (alpha + (beta + 3.0))) * (((1.0 + alpha) / (alpha + 2.0)) / (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = (1.0d0 / (alpha + (beta + 3.0d0))) * (((1.0d0 + alpha) / (alpha + 2.0d0)) / (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / ((beta + 4.0d0) + (alpha * 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = (1.0 / (alpha + (beta + 3.0))) * (((1.0 + alpha) / (alpha + 2.0)) / (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = (1.0 / (alpha + (beta + 3.0))) * (((1.0 + alpha) / (alpha + 2.0)) / (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = (1.0 / (alpha + (beta + 3.0))) * (((1.0 + alpha) / (alpha + 2.0)) / (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   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/ 99.3%

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

      2. associate-/r* 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l+ 95.5%

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

      5. associate-+r+ 95.5%

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

      6. *-commutative 95.5%

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

      7. distribute-rgt1-in 95.5%

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

      8. +-commutative 95.5%

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

      9. *-commutative 95.5%

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

      10. distribute-rgt1-in 95.5%

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

      11. +-commutative 95.5%

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

      12. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 98.3%

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

      1. associate-/r* 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. +-commutative 98.9%

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

      8. frac-times 98.3%

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

      9. *-un-lft-identity 98.3%

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

      10. +-commutative 98.3%

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

      11. +-commutative 98.3%

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

      12. +-commutative 98.3%

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

      13. +-commutative 98.3%

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

      14. +-commutative 98.3%

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

      15. +-commutative 98.3%

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

      16. associate-+r+ 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in beta around 0 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. +-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. times-frac 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 2.5 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-/r* 61.9%

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

      3. +-commutative 61.9%

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

      4. associate-+l+ 61.9%

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

      5. associate-+r+ 61.9%

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

      6. *-commutative 61.9%

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

      7. distribute-rgt1-in 61.9%

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

      8. +-commutative 61.9%

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

      9. *-commutative 61.9%

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

      10. distribute-rgt1-in 61.9%

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

      11. +-commutative 61.9%

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

      12. times-frac 89.8%

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

   3. Simplified 89.8%

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

      1. associate-*r/ 89.9%

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

      2. +-commutative 89.9%

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

   5. Applied egg-rr 89.9%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in beta around inf 87.0%

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

       1. associate-+r+ 87.0%

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

   12. Simplified 87.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + 2}}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

Alternative 4: 98.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. associate-/l/ 92.7%

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

   2. associate-/r* 84.1%

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

   3. +-commutative 84.1%

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

   4. associate-+l+ 84.1%

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

   5. associate-+r+ 84.1%

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

   6. *-commutative 84.1%

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

   7. distribute-rgt1-in 84.1%

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

   8. +-commutative 84.1%

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

   9. *-commutative 84.1%

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

   10. distribute-rgt1-in 84.1%

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

   11. +-commutative 84.1%

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

   12. times-frac 96.0%

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

3. Simplified 96.0%

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

   1. associate-*r/ 96.1%

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

   2. +-commutative 96.1%

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

5. Applied egg-rr 96.1%

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

   1. associate-*r/ 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. clear-num 99.8%

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

   5. frac-times 99.4%

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

   6. *-un-lft-identity 99.4%

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

   7. +-commutative 99.4%

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

   8. +-commutative 99.4%

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

   9. +-commutative 99.4%

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

   10. associate-+r+ 99.4%

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

   11. +-commutative 99.4%

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

   12. +-commutative 99.4%

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

   13. +-commutative 99.4%

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

   14. +-commutative 99.4%

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

   15. associate-+r+ 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in alpha around 0 70.6%

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

    1. associate-/l* 72.2%

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

    2. +-commutative 72.2%

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

    3. +-commutative 72.2%

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

12. Simplified 72.2%

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

13. Final simplification 72.2%

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

Alternative 5: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 68000000:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\frac{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 68000000.0)
   (/ (/ 1.0 (+ 2.0 beta)) (/ (* (+ 2.0 beta) (+ beta 3.0)) (+ 1.0 beta)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ beta 4.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 68000000.0) {
		tmp = (1.0 / (2.0 + beta)) / (((2.0 + beta) * (beta + 3.0)) / (1.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	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 <= 68000000.0d0) then
        tmp = (1.0d0 / (2.0d0 + beta)) / (((2.0d0 + beta) * (beta + 3.0d0)) / (1.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / (beta + 4.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 68000000.0) {
		tmp = (1.0 / (2.0 + beta)) / (((2.0 + beta) * (beta + 3.0)) / (1.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 68000000.0:
		tmp = (1.0 / (2.0 + beta)) / (((2.0 + beta) * (beta + 3.0)) / (1.0 + beta))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 68000000.0)
		tmp = Float64(Float64(1.0 / Float64(2.0 + beta)) / Float64(Float64(Float64(2.0 + beta) * Float64(beta + 3.0)) / Float64(1.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(beta + 4.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 68000000.0)
		tmp = (1.0 / (2.0 + beta)) / (((2.0 + beta) * (beta + 3.0)) / (1.0 + beta));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	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, 68000000.0], N[(N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.8e7

   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/ 99.2%

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

      2. associate-/r* 95.6%

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

      3. +-commutative 95.6%

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

      4. associate-+l+ 95.6%

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

      5. associate-+r+ 95.6%

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

      6. *-commutative 95.6%

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

      7. distribute-rgt1-in 95.6%

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

      8. +-commutative 95.6%

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

      9. *-commutative 95.6%

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

      10. distribute-rgt1-in 95.6%

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

      11. +-commutative 95.6%

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

      12. times-frac 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-/r* 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. clear-num 99.8%

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

      5. frac-times 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. +-commutative 99.3%

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

      8. +-commutative 99.3%

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

      9. +-commutative 99.3%

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

      10. associate-+r+ 99.3%

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

      11. +-commutative 99.3%

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

      12. +-commutative 99.3%

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

      13. +-commutative 99.3%

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

      14. +-commutative 99.3%

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

      15. associate-+r+ 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in alpha around 0 64.7%

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

   11. Taylor expanded in alpha around 0 64.7%

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

       1. +-commutative 3.3%

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

   13. Simplified 64.7%

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

   if 6.8e7 < beta

   1. Initial program 83.4%

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

      1. associate-/l/ 79.4%

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

      2. associate-/r* 60.6%

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

      3. +-commutative 60.6%

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

      4. associate-+l+ 60.6%

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

      5. associate-+r+ 60.6%

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

      6. *-commutative 60.6%

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

      7. distribute-rgt1-in 60.5%

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

      8. +-commutative 60.5%

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

      9. *-commutative 60.5%

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

      10. distribute-rgt1-in 60.6%

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

      11. +-commutative 60.6%

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

      12. times-frac 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r/ 89.5%

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

      2. +-commutative 89.5%

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

   5. Applied egg-rr 89.5%

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

      1. associate-*r/ 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in alpha around 0 82.8%

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

       1. associate-/l* 87.6%

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

       2. +-commutative 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

   13. Taylor expanded in beta around inf 87.6%

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

       1. +-commutative 87.6%

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

   15. Simplified 87.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 68000000:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\frac{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 65000000:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\frac{2 + \beta}{\frac{1 + \beta}{\beta + 3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 65000000.0)
   (/ (/ 1.0 (+ 2.0 beta)) (/ (+ 2.0 beta) (/ (+ 1.0 beta) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ beta 4.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 65000000.0) {
		tmp = (1.0 / (2.0 + beta)) / ((2.0 + beta) / ((1.0 + beta) / (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	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 <= 65000000.0d0) then
        tmp = (1.0d0 / (2.0d0 + beta)) / ((2.0d0 + beta) / ((1.0d0 + beta) / (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / (beta + 4.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 65000000.0) {
		tmp = (1.0 / (2.0 + beta)) / ((2.0 + beta) / ((1.0 + beta) / (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 65000000.0:
		tmp = (1.0 / (2.0 + beta)) / ((2.0 + beta) / ((1.0 + beta) / (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 65000000.0)
		tmp = Float64(Float64(1.0 / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) / Float64(Float64(1.0 + beta) / Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(beta + 4.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 65000000.0)
		tmp = (1.0 / (2.0 + beta)) / ((2.0 + beta) / ((1.0 + beta) / (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	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, 65000000.0], N[(N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] / N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5e7

   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/ 99.2%

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

      2. associate-/r* 95.6%

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

      3. +-commutative 95.6%

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

      4. associate-+l+ 95.6%

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

      5. associate-+r+ 95.6%

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

      6. *-commutative 95.6%

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

      7. distribute-rgt1-in 95.6%

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

      8. +-commutative 95.6%

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

      9. *-commutative 95.6%

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

      10. distribute-rgt1-in 95.6%

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

      11. +-commutative 95.6%

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

      12. times-frac 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-/r* 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. clear-num 99.8%

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

      5. frac-times 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. +-commutative 99.3%

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

      8. +-commutative 99.3%

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

      9. +-commutative 99.3%

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

      10. associate-+r+ 99.3%

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

      11. +-commutative 99.3%

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

      12. +-commutative 99.3%

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

      13. +-commutative 99.3%

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

      14. +-commutative 99.3%

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

      15. associate-+r+ 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in alpha around 0 64.7%

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

       1. associate-/l* 64.7%

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

       2. +-commutative 64.7%

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

       3. +-commutative 64.7%

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

   12. Simplified 64.7%

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

   13. Taylor expanded in alpha around 0 64.7%

       \[\leadsto \frac{\color{blue}{\frac{1}{2 + \beta}}}{\frac{\beta + 2}{\frac{1 + \beta}{\beta + 3}}} \]
   14. Step-by-step derivation

       1. +-commutative 3.3%

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

   15. Simplified 64.7%

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

   if 6.5e7 < beta

   1. Initial program 83.4%

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

      1. associate-/l/ 79.4%

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

      2. associate-/r* 60.6%

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

      3. +-commutative 60.6%

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

      4. associate-+l+ 60.6%

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

      5. associate-+r+ 60.6%

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

      6. *-commutative 60.6%

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

      7. distribute-rgt1-in 60.5%

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

      8. +-commutative 60.5%

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

      9. *-commutative 60.5%

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

      10. distribute-rgt1-in 60.6%

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

      11. +-commutative 60.6%

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

      12. times-frac 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r/ 89.5%

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

      2. +-commutative 89.5%

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

   5. Applied egg-rr 89.5%

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

      1. associate-*r/ 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in alpha around 0 82.8%

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

       1. associate-/l* 87.6%

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

       2. +-commutative 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

   13. Taylor expanded in beta around inf 87.6%

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

       1. +-commutative 87.6%

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

   15. Simplified 87.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 65000000:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\frac{2 + \beta}{\frac{1 + \beta}{\beta + 3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha (+ beta 3.0)) (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ beta 4.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.9d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + (beta + 3.0d0)) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / (beta + 4.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.9:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(beta + 4.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.9], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   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/ 99.3%

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

      2. associate-/r* 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l+ 95.5%

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

      5. associate-+r+ 95.5%

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

      6. *-commutative 95.5%

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

      7. distribute-rgt1-in 95.5%

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

      8. +-commutative 95.5%

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

      9. *-commutative 95.5%

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

      10. distribute-rgt1-in 95.5%

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

      11. +-commutative 95.5%

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

      12. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 98.3%

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

      1. associate-/r* 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. +-commutative 98.9%

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

      8. frac-times 98.3%

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

      9. *-un-lft-identity 98.3%

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

      10. +-commutative 98.3%

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

      11. +-commutative 98.3%

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

      12. +-commutative 98.3%

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

      13. +-commutative 98.3%

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

      14. +-commutative 98.3%

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

      15. +-commutative 98.3%

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

      16. associate-+r+ 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in beta around 0 98.9%

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

   if 2.89999999999999991 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-/r* 61.9%

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

      3. +-commutative 61.9%

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

      4. associate-+l+ 61.9%

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

      5. associate-+r+ 61.9%

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

      6. *-commutative 61.9%

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

      7. distribute-rgt1-in 61.9%

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

      8. +-commutative 61.9%

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

      9. *-commutative 61.9%

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

      10. distribute-rgt1-in 61.9%

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

      11. +-commutative 61.9%

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

      12. times-frac 89.8%

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

   3. Simplified 89.8%

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

      1. associate-*r/ 89.9%

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

      2. +-commutative 89.9%

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

   5. Applied egg-rr 89.9%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in alpha around 0 83.4%

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

       1. associate-/l* 88.0%

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

       2. +-commutative 88.0%

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

       3. +-commutative 88.0%

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

   12. Simplified 88.0%

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

   13. Taylor expanded in beta around inf 86.7%

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

       1. +-commutative 86.7%

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

   15. Simplified 86.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \]

Alternative 8: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha (+ beta 3.0)) (+ alpha 2.0)))
   (/
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ (+ beta 4.0) (* alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + (beta + 3.0d0)) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / ((beta + 4.0d0) + (alpha * 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   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/ 99.3%

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

      2. associate-/r* 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l+ 95.5%

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

      5. associate-+r+ 95.5%

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

      6. *-commutative 95.5%

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

      7. distribute-rgt1-in 95.5%

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

      8. +-commutative 95.5%

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

      9. *-commutative 95.5%

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

      10. distribute-rgt1-in 95.5%

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

      11. +-commutative 95.5%

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

      12. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 98.3%

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

      1. associate-/r* 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. +-commutative 98.9%

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

      8. frac-times 98.3%

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

      9. *-un-lft-identity 98.3%

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

      10. +-commutative 98.3%

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

      11. +-commutative 98.3%

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

      12. +-commutative 98.3%

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

      13. +-commutative 98.3%

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

      14. +-commutative 98.3%

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

      15. +-commutative 98.3%

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

      16. associate-+r+ 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in beta around 0 98.9%

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

   if 5.20000000000000018 < beta

   1. Initial program 84.0%

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

      1. associate-/l/ 80.0%

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

      2. associate-/r* 61.9%

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

      3. +-commutative 61.9%

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

      4. associate-+l+ 61.9%

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

      5. associate-+r+ 61.9%

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

      6. *-commutative 61.9%

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

      7. distribute-rgt1-in 61.9%

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

      8. +-commutative 61.9%

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

      9. *-commutative 61.9%

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

      10. distribute-rgt1-in 61.9%

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

      11. +-commutative 61.9%

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

      12. times-frac 89.8%

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

   3. Simplified 89.8%

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

      1. associate-*r/ 89.9%

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

      2. +-commutative 89.9%

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

   5. Applied egg-rr 89.9%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in beta around inf 87.0%

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

       1. associate-+r+ 87.0%

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

   12. Simplified 87.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

Alternative 9: 98.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3)
   (/ (+ 0.5 (* alpha 0.25)) (* (+ alpha (+ beta 3.0)) (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ beta 4.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = (0.5 + (alpha * 0.25)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	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 <= 1.3d0) then
        tmp = (0.5d0 + (alpha * 0.25d0)) / ((alpha + (beta + 3.0d0)) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / (beta + 4.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = (0.5 + (alpha * 0.25)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.3:
		tmp = (0.5 + (alpha * 0.25)) / ((alpha + (beta + 3.0)) * (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3)
		tmp = Float64(Float64(0.5 + Float64(alpha * 0.25)) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(beta + 4.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3)
		tmp = (0.5 + (alpha * 0.25)) / ((alpha + (beta + 3.0)) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	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, 1.3], N[(N[(0.5 + N[(alpha * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.30000000000000004

   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/ 99.2%

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

      2. associate-/r* 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l+ 95.5%

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

      5. associate-+r+ 95.5%

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

      6. *-commutative 95.5%

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

      7. distribute-rgt1-in 95.5%

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

      8. +-commutative 95.5%

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

      9. *-commutative 95.5%

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

      10. distribute-rgt1-in 95.5%

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

      11. +-commutative 95.5%

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

      12. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 98.3%

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

      1. associate-/r* 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. +-commutative 98.9%

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

      8. frac-times 98.3%

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

      9. *-un-lft-identity 98.3%

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

      10. +-commutative 98.3%

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

      11. +-commutative 98.3%

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

      12. +-commutative 98.3%

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

      13. +-commutative 98.3%

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

      14. +-commutative 98.3%

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

      15. +-commutative 98.3%

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

      16. associate-+r+ 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in beta around 0 98.9%

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

   8. Taylor expanded in alpha around 0 82.4%

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

      1. *-commutative 82.4%

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

   10. Simplified 82.4%

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

   if 1.30000000000000004 < beta

   1. Initial program 84.1%

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

      1. associate-/l/ 80.3%

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

      2. associate-/r* 62.3%

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

      3. +-commutative 62.3%

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

      4. associate-+l+ 62.3%

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

      5. associate-+r+ 62.3%

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

      6. *-commutative 62.3%

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

      7. distribute-rgt1-in 62.3%

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

      8. +-commutative 62.3%

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

      9. *-commutative 62.3%

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

      10. distribute-rgt1-in 62.3%

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

      11. +-commutative 62.3%

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

      12. times-frac 89.9%

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

   3. Simplified 89.9%

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

      1. associate-*r/ 90.0%

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

      2. +-commutative 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. associate-*r/ 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in alpha around 0 82.4%

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

       1. associate-/l* 87.1%

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

       2. +-commutative 87.1%

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

       3. +-commutative 87.1%

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

   12. Simplified 87.1%

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

   13. Taylor expanded in beta around inf 85.8%

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

       1. +-commutative 85.8%

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

   15. Simplified 85.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3:\\ \;\;\;\;\frac{0.5 + \alpha \cdot 0.25}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \]

Alternative 10: 97.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (/ 0.25 (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ beta 4.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	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 <= 1.2d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / (beta + 4.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(beta + 4.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (beta + 4.0);
	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, 1.2], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   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/ 99.2%

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

      2. associate-/r* 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l+ 95.5%

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

      5. associate-+r+ 95.5%

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

      6. *-commutative 95.5%

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

      7. distribute-rgt1-in 95.5%

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

      8. +-commutative 95.5%

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

      9. *-commutative 95.5%

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

      10. distribute-rgt1-in 95.5%

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

      11. +-commutative 95.5%

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

      12. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 98.3%

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

      1. associate-/r* 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. +-commutative 98.9%

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

      5. associate-+r+ 98.9%

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

      6. +-commutative 98.9%

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

      7. +-commutative 98.9%

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

      8. frac-times 98.3%

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

      9. *-un-lft-identity 98.3%

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

      10. +-commutative 98.3%

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

      11. +-commutative 98.3%

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

      12. +-commutative 98.3%

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

      13. +-commutative 98.3%

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

      14. +-commutative 98.3%

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

      15. +-commutative 98.3%

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

      16. associate-+r+ 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in beta around 0 98.9%

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

   8. Taylor expanded in alpha around 0 64.2%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 64.2%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   10. Simplified 64.2%

       \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 1.19999999999999996 < beta

   1. Initial program 84.1%

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

      1. associate-/l/ 80.3%

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

      2. associate-/r* 62.3%

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

      3. +-commutative 62.3%

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

      4. associate-+l+ 62.3%

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

      5. associate-+r+ 62.3%

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

      6. *-commutative 62.3%

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

      7. distribute-rgt1-in 62.3%

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

      8. +-commutative 62.3%

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

      9. *-commutative 62.3%

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

      10. distribute-rgt1-in 62.3%

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

      11. +-commutative 62.3%

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

      12. times-frac 89.9%

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

   3. Simplified 89.9%

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

      1. associate-*r/ 90.0%

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

      2. +-commutative 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. associate-*r/ 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in alpha around 0 82.4%

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

       1. associate-/l* 87.1%

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

       2. +-commutative 87.1%

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

       3. +-commutative 87.1%

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

   12. Simplified 87.1%

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

   13. Taylor expanded in beta around inf 85.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{4 + \beta}} \]
   14. Step-by-step derivation

       1. +-commutative 85.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta + 4}} \]

   15. Simplified 85.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta + 4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta + 4}\\ \end{array} \]

Alternative 11: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ 0.25 (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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/ 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 95.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 95.5%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 95.5%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 95.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 95.5%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 95.5%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 99.2%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around 0 98.3%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

      2. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \]

      4. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right)} + \alpha}}{\alpha + \left(\beta + 3\right)} \]

      5. associate-+r+ 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

      6. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\beta + \alpha\right)} + 3} \]

      7. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\beta + \alpha\right)}} \]

      8. frac-times 98.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      9. *-un-lft-identity 98.3%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      14. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      15. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      16. associate-+r+ 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   7. Taylor expanded in beta around 0 98.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Taylor expanded in alpha around 0 63.8%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 63.8%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4.5 < beta

   1. Initial program 84.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 61.9%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 61.9%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 61.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 61.9%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 61.9%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 89.8%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.8%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. *-commutative 89.8%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      7. associate-+r+ 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\alpha + \beta\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) + \alpha}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\left(2 + \beta\right) + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      3. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\beta + \left(2 + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      4. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta}}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      5. frac-times 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      6. *-un-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\left(\beta + 2\right) + \alpha}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\alpha + \left(\beta + 2\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \color{blue}{\left(2 + \beta\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      15. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   10. Taylor expanded in beta around inf 85.4%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta}\\ \end{array} \]

Alternative 12: 91.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5) (/ 0.25 (+ beta 3.0)) (/ 1.0 (* beta (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = 1.0 / (beta * (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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/ 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 95.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 95.5%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 95.5%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 95.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 95.5%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 95.5%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 99.2%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around 0 98.3%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

      2. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \]

      4. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right)} + \alpha}}{\alpha + \left(\beta + 3\right)} \]

      5. associate-+r+ 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

      6. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\beta + \alpha\right)} + 3} \]

      7. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\beta + \alpha\right)}} \]

      8. frac-times 98.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      9. *-un-lft-identity 98.3%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      14. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      15. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      16. associate-+r+ 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   7. Taylor expanded in beta around 0 98.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Taylor expanded in alpha around 0 63.8%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 63.8%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4.5 < beta

   1. Initial program 84.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 61.9%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 61.9%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 61.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 61.9%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 61.9%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 89.8%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.8%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. *-commutative 89.8%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      7. associate-+r+ 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\alpha + \beta\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) + \alpha}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\left(2 + \beta\right) + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      3. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\beta + \left(2 + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      4. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta}}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      5. frac-times 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      6. *-un-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\left(\beta + 2\right) + \alpha}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\alpha + \left(\beta + 2\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \color{blue}{\left(2 + \beta\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      15. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   10. Taylor expanded in beta around inf 85.4%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta}} \]

   11. Taylor expanded in alpha around 0 76.8%

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(2 + \beta\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 76.8%

          \[\leadsto \frac{1}{\beta \cdot \color{blue}{\left(\beta + 2\right)}} \]

   13. Simplified 76.8%

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(\beta + 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \]

Alternative 13: 91.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5) (/ 0.25 (+ beta 3.0)) (/ (/ 1.0 (+ 2.0 beta)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / (2.0 + beta)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = (1.0d0 / (2.0d0 + beta)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / (2.0 + beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = (1.0 / (2.0 + beta)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(2.0 + beta)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = (1.0 / (2.0 + beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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/ 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 95.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 95.5%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 95.5%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 95.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 95.5%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 95.5%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 99.2%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around 0 98.3%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

      2. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \]

      4. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right)} + \alpha}}{\alpha + \left(\beta + 3\right)} \]

      5. associate-+r+ 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

      6. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\beta + \alpha\right)} + 3} \]

      7. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\beta + \alpha\right)}} \]

      8. frac-times 98.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      9. *-un-lft-identity 98.3%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      14. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      15. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      16. associate-+r+ 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   7. Taylor expanded in beta around 0 98.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Taylor expanded in alpha around 0 63.8%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 63.8%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4.5 < beta

   1. Initial program 84.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 61.9%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 61.9%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 61.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 61.9%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 61.9%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 89.8%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.8%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. *-commutative 89.8%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      7. associate-+r+ 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\alpha + \beta\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) + \alpha}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\left(2 + \beta\right) + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      3. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\beta + \left(2 + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      4. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta}}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      5. frac-times 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      6. *-un-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\left(\beta + 2\right) + \alpha}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\alpha + \left(\beta + 2\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \color{blue}{\left(2 + \beta\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      15. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   10. Taylor expanded in beta around inf 85.4%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta}} \]

   11. Taylor expanded in alpha around 0 77.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{2 + \beta}}}{\beta} \]
   12. Step-by-step derivation

       1. +-commutative 77.3%

          \[\leadsto \frac{\frac{1}{\color{blue}{\beta + 2}}}{\beta} \]

   13. Simplified 77.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{\beta + 2}}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\beta}\\ \end{array} \]

Alternative 14: 96.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) (/ 0.25 (+ beta 3.0)) (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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/ 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 95.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 95.5%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 95.5%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 95.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 95.5%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 95.5%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 99.2%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around 0 98.3%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

      2. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \]

      4. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right)} + \alpha}}{\alpha + \left(\beta + 3\right)} \]

      5. associate-+r+ 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

      6. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\beta + \alpha\right)} + 3} \]

      7. +-commutative 98.9%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\beta + \alpha\right)}} \]

      8. frac-times 98.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      9. *-un-lft-identity 98.3%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      14. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      15. +-commutative 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      16. associate-+r+ 98.3%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   7. Taylor expanded in beta around 0 98.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Taylor expanded in alpha around 0 63.8%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 63.8%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 6 < beta

   1. Initial program 84.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 61.9%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 61.9%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 61.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 61.9%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 61.9%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 89.8%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.8%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. *-commutative 89.8%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      7. associate-+r+ 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\alpha + \beta\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) + \alpha}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\left(2 + \beta\right) + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      3. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\beta + \left(2 + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      4. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta}}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      5. frac-times 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      6. *-un-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\left(\beta + 2\right) + \alpha}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\alpha + \left(\beta + 2\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \color{blue}{\left(2 + \beta\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      15. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   10. Taylor expanded in beta around inf 85.4%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta}} \]

   11. Taylor expanded in beta around inf 85.3%

       \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 15: 47.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 11.5:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 11.5)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 11.5) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 11.5d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 11.5) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 11.5:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 11.5)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 11.5)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 11.5], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 11.5

   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/ 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 95.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 95.5%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 95.5%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 95.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 95.5%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 95.5%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 99.2%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around 0 98.3%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Taylor expanded in alpha around 0 64.4%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{1}{\left(2 + \beta\right) \cdot \color{blue}{\left(\beta + 3\right)}} \]

   7. Simplified 64.4%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0 64.4%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   9. Taylor expanded in alpha around 0 62.6%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   10. Step-by-step derivation

       1. *-commutative 62.6%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   11. Simplified 62.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 11.5 < beta

   1. Initial program 84.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 61.9%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 61.9%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 61.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 61.9%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 61.9%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 89.8%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.8%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. *-commutative 89.8%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      7. associate-+r+ 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\alpha + \beta\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) + \alpha}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\left(2 + \beta\right) + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      3. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\beta + \left(2 + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      4. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta}}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      5. frac-times 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      6. *-un-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\left(\beta + 2\right) + \alpha}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\alpha + \left(\beta + 2\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \color{blue}{\left(2 + \beta\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      15. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   10. Taylor expanded in beta around inf 85.4%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta}} \]

   11. Taylor expanded in alpha around inf 7.0%

       \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 11.5:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 16: 47.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   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/ 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 95.5%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 95.5%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 95.5%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 95.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 95.5%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 95.5%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 95.5%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 99.2%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around 0 98.3%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Taylor expanded in alpha around 0 64.4%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{1}{\left(2 + \beta\right) \cdot \color{blue}{\left(\beta + 3\right)}} \]

   7. Simplified 64.4%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0 64.4%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   9. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 12 < beta

   1. Initial program 84.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 61.9%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+l+ 61.9%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. associate-+r+ 61.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. distribute-rgt1-in 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. +-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. *-commutative 61.9%

         \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. distribute-rgt1-in 61.9%

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. +-commutative 61.9%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. times-frac 89.8%

          \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.8%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. *-commutative 89.8%

         \[\leadsto \color{blue}{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      7. associate-+r+ 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\alpha + \beta\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) + \alpha}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \cdot \frac{1 + \beta}{\left(2 + \beta\right) + \alpha}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      3. associate-+r+ 99.7%

         \[\leadsto \frac{1 + \beta}{\color{blue}{\beta + \left(2 + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      4. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta}}} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{3 + \left(\beta + \alpha\right)} \]

      5. frac-times 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      6. *-un-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\left(\beta + 2\right) + \alpha}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\alpha + \left(\beta + 2\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      12. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \color{blue}{\left(2 + \beta\right)}}{1 + \beta} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      13. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      15. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   10. Taylor expanded in beta around inf 85.4%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\beta}} \]

   11. Taylor expanded in alpha around inf 7.0%

       \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 17: 45.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\alpha + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ alpha 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (alpha + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (alpha + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(alpha + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (alpha + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 92.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-/r* 84.1%

      \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

   3. +-commutative 84.1%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   4. associate-+l+ 84.1%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   5. associate-+r+ 84.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   6. *-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   8. +-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   9. *-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   10. distribute-rgt1-in 84.1%

       \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   11. +-commutative 84.1%

       \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   12. times-frac 96.0%

       \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 96.0%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around 0 81.8%

   \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

5. Taylor expanded in alpha around 0 59.4%

   \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
6. Step-by-step derivation

   1. +-commutative 59.4%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{1}{\left(2 + \beta\right) \cdot \color{blue}{\left(\beta + 3\right)}} \]

7. Simplified 59.4%

   \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}} \]

8. Taylor expanded in beta around 0 44.2%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

9. Final simplification 44.2%

   \[\leadsto \frac{0.16666666666666666}{\alpha + 2} \]

Alternative 18: 47.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (beta + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 92.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-/r* 84.1%

      \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

   3. +-commutative 84.1%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   4. associate-+l+ 84.1%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   5. associate-+r+ 84.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   6. *-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   8. +-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   9. *-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   10. distribute-rgt1-in 84.1%

       \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   11. +-commutative 84.1%

       \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   12. times-frac 96.0%

       \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 96.0%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around 0 81.8%

   \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
5. Step-by-step derivation

   1. associate-/r* 82.3%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

   2. +-commutative 82.3%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

   3. +-commutative 82.3%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \]

   4. +-commutative 82.3%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right)} + \alpha}}{\alpha + \left(\beta + 3\right)} \]

   5. associate-+r+ 82.3%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

   6. +-commutative 82.3%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{\left(\beta + \alpha\right)} + 3} \]

   7. +-commutative 82.3%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\color{blue}{3 + \left(\beta + \alpha\right)}} \]

   8. frac-times 81.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

   9. *-un-lft-identity 81.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \beta\right) + \alpha}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   10. +-commutative 81.9%

       \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   11. +-commutative 81.9%

       \[\leadsto \frac{\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   12. +-commutative 81.9%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\left(2 + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   13. +-commutative 81.9%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   14. +-commutative 81.9%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}} \]

   15. +-commutative 81.9%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

   16. associate-+r+ 81.9%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}} \]

6. Applied egg-rr 81.9%

   \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

7. Taylor expanded in beta around 0 71.0%

   \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\alpha + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

8. Taylor expanded in alpha around 0 44.5%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
9. Step-by-step derivation

   1. +-commutative 44.5%

      \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

10. Simplified 44.5%

    \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

11. Final simplification 44.5%

    \[\leadsto \frac{0.25}{\beta + 3} \]

Alternative 19: 45.5% 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 94.5%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 92.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-/r* 84.1%

      \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

   3. +-commutative 84.1%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   4. associate-+l+ 84.1%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   5. associate-+r+ 84.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   6. *-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(\alpha + 1\right) \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   8. +-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)} \cdot \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   9. *-commutative 84.1%

      \[\leadsto \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   10. distribute-rgt1-in 84.1%

       \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   11. +-commutative 84.1%

       \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   12. times-frac 96.0%

       \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 96.0%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in beta around 0 81.8%

   \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

5. Taylor expanded in alpha around 0 59.4%

   \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
6. Step-by-step derivation

   1. +-commutative 59.4%

      \[\leadsto \frac{1}{2 + \alpha} \cdot \frac{1}{\left(2 + \beta\right) \cdot \color{blue}{\left(\beta + 3\right)}} \]

7. Simplified 59.4%

   \[\leadsto \frac{1}{2 + \alpha} \cdot \color{blue}{\frac{1}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}} \]

8. Taylor expanded in beta around 0 44.2%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

9. Taylor expanded in alpha around 0 43.2%

   \[\leadsto \color{blue}{0.08333333333333333} \]

10. Final simplification 43.2%

    \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023309 
(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)))