Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.8%

Time: 22.7s

Alternatives: 18

Speedup: 2.9×

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: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 83.5%

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

   1. associate-+r+ 83.5%

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

   2. fma-undefine 83.5%

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

   3. *-commutative 83.5%

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

   4. associate-+l+ 83.5%

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

   5. +-commutative 83.5%

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

   6. associate-+l+ 83.5%

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

   7. *-commutative 83.5%

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

   8. associate-*r* 83.5%

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

   9. associate-+r+ 83.5%

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

   10. +-commutative 83.5%

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

   11. associate-/l/ 92.8%

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

   12. clear-num 92.9%

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

   13. inv-pow 92.9%

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

6. Applied egg-rr 92.9%

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

   1. unpow-1 92.9%

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

   2. associate-/l* 93.7%

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

   3. associate-+r+ 93.7%

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

   4. +-commutative 93.7%

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

   5. +-commutative 93.7%

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

   6. associate-+r+ 93.7%

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

   7. +-commutative 93.7%

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

   8. fma-undefine 93.7%

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

   9. +-commutative 93.7%

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

   10. *-commutative 93.7%

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

   11. +-commutative 93.7%

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

   12. associate-+r+ 93.7%

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

   13. distribute-rgt1-in 93.7%

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

   14. +-commutative 93.7%

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

   15. associate-+r+ 93.7%

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

   16. +-commutative 93.7%

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

   17. +-commutative 93.7%

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

8. Simplified 93.7%

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

   1. associate-*r/ 92.9%

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

   2. associate-+l+ 92.9%

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

   3. +-commutative 92.9%

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

   4. clear-num 92.9%

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

   5. *-un-lft-identity 92.9%

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

   6. associate-/l* 96.4%

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

   7. associate-+r+ 96.4%

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

   8. +-commutative 96.4%

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

   9. +-commutative 96.4%

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

   10. associate-+r+ 96.4%

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

   11. +-commutative 96.4%

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

   12. +-commutative 96.4%

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

   13. +-commutative 96.4%

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

10. Applied egg-rr 96.4%

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

    1. *-lft-identity 96.4%

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

    2. times-frac 99.7%

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

    3. +-commutative 99.7%

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

    4. +-commutative 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.1)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ 3.0 (+ beta alpha))))
   (*
    (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ beta (+ alpha 3.0)))
    (+ 1.0 (/ (- -1.0 alpha) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.1d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (3.0d0 + (beta + alpha)))
    else
        tmp = (((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (beta + (alpha + 3.0d0))) * (1.0d0 + (((-1.0d0) - alpha) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)));
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.1:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)))
	else:
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.1)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(beta + Float64(alpha + 3.0))) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.1)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (beta + alpha)));
	else
		tmp = (((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + (alpha + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.1], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000009

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-+l+ 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around 0 98.5%

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

   6. Taylor expanded in beta around 0 99.0%

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

   if 3.10000000000000009 < beta

   1. Initial program 82.6%

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

   3. Simplified 64.4%

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

      1. associate-+r+ 64.4%

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

      2. fma-undefine 64.4%

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

      3. *-commutative 64.4%

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

      4. associate-+l+ 64.4%

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

      5. +-commutative 64.4%

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

      6. associate-+l+ 64.4%

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

      7. *-commutative 64.4%

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

      8. associate-*r* 64.4%

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

      9. associate-+r+ 64.4%

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

      10. +-commutative 64.4%

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

      11. associate-/l/ 79.3%

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

      12. clear-num 79.3%

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

      13. inv-pow 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. unpow-1 79.3%

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

      2. associate-/l* 81.7%

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

      3. associate-+r+ 81.7%

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

      4. +-commutative 81.7%

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

      5. +-commutative 81.7%

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

      6. associate-+r+ 81.7%

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

      7. +-commutative 81.7%

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

      8. fma-undefine 81.7%

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

      9. +-commutative 81.7%

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

      10. *-commutative 81.7%

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

      11. +-commutative 81.7%

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

      12. associate-+r+ 81.7%

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

      13. distribute-rgt1-in 81.7%

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

      14. +-commutative 81.7%

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

      15. associate-+r+ 81.7%

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

      16. +-commutative 81.7%

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

      17. +-commutative 81.7%

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

   8. Simplified 81.7%

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

      1. associate-*r/ 79.3%

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

      2. associate-+l+ 79.3%

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

      3. +-commutative 79.3%

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

      4. clear-num 79.4%

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

      5. *-un-lft-identity 79.4%

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

      6. associate-/l* 89.8%

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

      7. associate-+r+ 89.8%

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

      8. +-commutative 89.8%

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

      9. +-commutative 89.8%

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

      10. associate-+r+ 89.8%

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

      11. +-commutative 89.8%

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

      12. +-commutative 89.8%

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

      13. +-commutative 89.8%

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

   10. Applied egg-rr 89.8%

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

       1. *-lft-identity 89.8%

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

       2. times-frac 99.7%

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

       3. +-commutative 99.7%

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

       4. +-commutative 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in beta around inf 84.2%

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

       1. mul-1-neg 82.8%

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

       2. unsub-neg 82.8%

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

   15. Simplified 84.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \frac{\beta}{t\_0}}{t\_0}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.3e+20)
     (/
      1.0
      (*
       (+ 2.0 (+ beta alpha))
       (/ (* (+ beta 2.0) (+ beta 3.0)) (+ 1.0 beta))))
     (/ (/ (* (+ 1.0 alpha) (/ beta t_0)) t_0) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.3e+20) {
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.3d+20) then
        tmp = 1.0d0 / ((2.0d0 + (beta + alpha)) * (((beta + 2.0d0) * (beta + 3.0d0)) / (1.0d0 + beta)))
    else
        tmp = (((1.0d0 + alpha) * (beta / t_0)) / t_0) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.3e+20) {
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.3e+20:
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)))
	else:
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.3e+20)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(beta + alpha)) * Float64(Float64(Float64(beta + 2.0) * Float64(beta + 3.0)) / Float64(1.0 + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(beta / t_0)) / t_0) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.3e+20)
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	else
		tmp = (((1.0 + alpha) * (beta / t_0)) / t_0) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.3e+20], N[(1.0 / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3e20

   1. Initial program 99.9%

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

   3. Simplified 93.4%

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

      1. associate-+r+ 93.4%

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

      2. fma-undefine 93.4%

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

      3. *-commutative 93.4%

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

      4. associate-+l+ 93.4%

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

      5. +-commutative 93.4%

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

      6. associate-+l+ 93.4%

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

      7. *-commutative 93.4%

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

      8. associate-*r* 93.4%

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

      9. associate-+r+ 93.4%

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

      10. +-commutative 93.4%

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

      11. associate-/l/ 99.7%

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

      12. clear-num 99.7%

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

      13. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-undefine 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+r+ 99.7%

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

      13. distribute-rgt1-in 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 64.2%

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

   if 2.3e20 < beta

   1. Initial program 81.5%

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

   3. Simplified 62.2%

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

   6. Applied egg-rr 78.1%

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

      1. associate-*r/ 78.1%

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

      2. *-rgt-identity 78.1%

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

      3. fma-undefine 78.1%

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

      4. +-commutative 78.1%

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

      5. *-commutative 78.1%

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

      6. +-commutative 78.1%

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

      7. associate-+r+ 78.1%

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

      8. distribute-rgt1-in 78.1%

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

      9. +-commutative 78.1%

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

      10. associate-+r+ 78.1%

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

      11. +-commutative 78.1%

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

      12. +-commutative 78.1%

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

      13. associate-+r+ 78.1%

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

      14. +-commutative 78.1%

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

      15. +-commutative 78.1%

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

      16. +-commutative 78.1%

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

      17. +-commutative 78.1%

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

      18. +-commutative 78.1%

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

      19. associate-+l+ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in beta around inf 78.1%

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

       1. *-commutative 78.1%

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

   11. Simplified 78.1%

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

       1. *-un-lft-identity 78.1%

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

       2. associate-/r* 81.6%

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

       3. associate-/l* 99.8%

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

       4. associate-+r+ 99.8%

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

       5. +-commutative 99.8%

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

       6. associate-+r+ 99.8%

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

       7. +-commutative 99.8%

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

       8. +-commutative 99.8%

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

   13. Applied egg-rr 99.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.3e+15)
   (/ (+ 1.0 beta) (* (+ 2.0 (+ beta alpha)) (* (+ beta 2.0) (+ beta 3.0))))
   (*
    (+ 1.0 (/ (- -1.0 alpha) beta))
    (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3e+15) {
		tmp = (1.0 + beta) / ((2.0 + (beta + alpha)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.3d+15) then
        tmp = (1.0d0 + beta) / ((2.0d0 + (beta + alpha)) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = (1.0d0 + (((-1.0d0) - alpha) / beta)) * (((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3e+15) {
		tmp = (1.0 + beta) / ((2.0 + (beta + alpha)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.3e+15:
		tmp = (1.0 + beta) / ((2.0 + (beta + alpha)) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.3e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + Float64(beta + alpha)) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) * Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.3e+15)
		tmp = (1.0 + beta) / ((2.0 + (beta + alpha)) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.3e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.3e15

   1. Initial program 99.9%

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

   3. Simplified 93.3%

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

      1. associate-+r+ 93.3%

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

      2. fma-undefine 93.3%

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

      3. *-commutative 93.3%

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

      4. associate-+l+ 93.3%

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

      5. +-commutative 93.3%

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

      6. associate-+l+ 93.3%

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

      7. *-commutative 93.3%

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

      8. associate-*r* 93.3%

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

      9. associate-+r+ 93.3%

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

      10. +-commutative 93.3%

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

      11. associate-/l/ 99.7%

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

      12. clear-num 99.7%

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

      13. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-undefine 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+r+ 99.7%

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

      13. distribute-rgt1-in 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. associate-/l* 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. times-frac 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around 0 64.4%

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

       1. associate-/r* 64.4%

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

       2. +-commutative 64.4%

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

   15. Simplified 64.4%

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

       1. *-commutative 64.4%

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

       2. associate-/l/ 64.4%

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

       3. frac-times 64.4%

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

       4. *-un-lft-identity 64.4%

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

       5. +-commutative 64.4%

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

       6. associate-+r+ 64.4%

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

   17. Applied egg-rr 64.4%

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

   if 4.3e15 < beta

   1. Initial program 81.8%

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

   3. Simplified 62.6%

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

      1. associate-+r+ 62.6%

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

      2. fma-undefine 62.6%

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

      3. *-commutative 62.6%

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

      4. associate-+l+ 62.6%

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

      5. +-commutative 62.6%

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

      6. associate-+l+ 62.6%

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

      7. *-commutative 62.6%

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

      8. associate-*r* 62.6%

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

      9. associate-+r+ 62.6%

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

      10. +-commutative 62.6%

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

      11. associate-/l/ 78.4%

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

      12. clear-num 78.3%

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

      13. inv-pow 78.3%

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

   6. Applied egg-rr 78.4%

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

      1. unpow-1 78.4%

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

      2. associate-/l* 80.8%

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

      3. associate-+r+ 80.8%

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

      4. +-commutative 80.8%

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

      5. +-commutative 80.8%

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

      6. associate-+r+ 80.8%

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

      7. +-commutative 80.8%

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

      8. fma-undefine 80.8%

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

      9. +-commutative 80.8%

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

      10. *-commutative 80.8%

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

      11. +-commutative 80.8%

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

      12. associate-+r+ 80.8%

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

      13. distribute-rgt1-in 80.8%

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

      14. +-commutative 80.8%

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

      15. associate-+r+ 80.8%

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

      16. +-commutative 80.8%

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

      17. +-commutative 80.8%

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

   8. Simplified 80.8%

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

      1. associate-*r/ 78.4%

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

      2. associate-+l+ 78.4%

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

      3. +-commutative 78.4%

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

      4. clear-num 78.4%

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

      5. *-un-lft-identity 78.4%

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

      6. associate-/l* 89.3%

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

      7. associate-+r+ 89.3%

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

      8. +-commutative 89.3%

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

      9. +-commutative 89.3%

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

      10. associate-+r+ 89.3%

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

      11. +-commutative 89.3%

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

      12. +-commutative 89.3%

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

      13. +-commutative 89.3%

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

   10. Applied egg-rr 89.3%

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

       1. *-lft-identity 89.3%

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

       2. times-frac 99.7%

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

       3. +-commutative 99.7%

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

       4. +-commutative 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in beta around inf 86.1%

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

   14. Taylor expanded in beta around inf 85.3%

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

       1. mul-1-neg 85.3%

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

       2. unsub-neg 85.3%

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

   16. Simplified 85.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3e+15)
   (*
    (/ (+ 1.0 beta) (+ alpha (+ beta 2.0)))
    (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0)))
   (*
    (+ 1.0 (/ (- -1.0 alpha) beta))
    (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3e+15) {
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d+15) then
        tmp = ((1.0d0 + beta) / (alpha + (beta + 2.0d0))) * ((1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0))
    else
        tmp = (1.0d0 + (((-1.0d0) - alpha) / beta)) * (((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3e+15) {
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3e+15:
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * ((1.0 / (beta + 2.0)) / (beta + 3.0))
	else:
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) * Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3e+15)
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	else
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3e15

   1. Initial program 99.9%

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

   3. Simplified 93.3%

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

      1. associate-+r+ 93.3%

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

      2. fma-undefine 93.3%

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

      3. *-commutative 93.3%

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

      4. associate-+l+ 93.3%

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

      5. +-commutative 93.3%

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

      6. associate-+l+ 93.3%

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

      7. *-commutative 93.3%

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

      8. associate-*r* 93.3%

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

      9. associate-+r+ 93.3%

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

      10. +-commutative 93.3%

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

      11. associate-/l/ 99.7%

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

      12. clear-num 99.7%

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

      13. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-undefine 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+r+ 99.7%

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

      13. distribute-rgt1-in 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. associate-/l* 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. times-frac 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around 0 64.4%

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

       1. associate-/r* 64.4%

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

       2. +-commutative 64.4%

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

   15. Simplified 64.4%

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

   if 2.3e15 < beta

   1. Initial program 81.8%

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

   3. Simplified 62.6%

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

      1. associate-+r+ 62.6%

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

      2. fma-undefine 62.6%

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

      3. *-commutative 62.6%

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

      4. associate-+l+ 62.6%

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

      5. +-commutative 62.6%

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

      6. associate-+l+ 62.6%

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

      7. *-commutative 62.6%

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

      8. associate-*r* 62.6%

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

      9. associate-+r+ 62.6%

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

      10. +-commutative 62.6%

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

      11. associate-/l/ 78.4%

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

      12. clear-num 78.3%

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

      13. inv-pow 78.3%

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

   6. Applied egg-rr 78.4%

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

      1. unpow-1 78.4%

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

      2. associate-/l* 80.8%

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

      3. associate-+r+ 80.8%

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

      4. +-commutative 80.8%

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

      5. +-commutative 80.8%

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

      6. associate-+r+ 80.8%

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

      7. +-commutative 80.8%

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

      8. fma-undefine 80.8%

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

      9. +-commutative 80.8%

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

      10. *-commutative 80.8%

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

      11. +-commutative 80.8%

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

      12. associate-+r+ 80.8%

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

      13. distribute-rgt1-in 80.8%

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

      14. +-commutative 80.8%

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

      15. associate-+r+ 80.8%

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

      16. +-commutative 80.8%

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

      17. +-commutative 80.8%

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

   8. Simplified 80.8%

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

      1. associate-*r/ 78.4%

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

      2. associate-+l+ 78.4%

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

      3. +-commutative 78.4%

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

      4. clear-num 78.4%

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

      5. *-un-lft-identity 78.4%

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

      6. associate-/l* 89.3%

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

      7. associate-+r+ 89.3%

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

      8. +-commutative 89.3%

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

      9. +-commutative 89.3%

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

      10. associate-+r+ 89.3%

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

      11. +-commutative 89.3%

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

      12. +-commutative 89.3%

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

      13. +-commutative 89.3%

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

   10. Applied egg-rr 89.3%

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

       1. *-lft-identity 89.3%

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

       2. times-frac 99.7%

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

       3. +-commutative 99.7%

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

       4. +-commutative 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in beta around inf 86.1%

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

   14. Taylor expanded in beta around inf 85.3%

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

       1. mul-1-neg 85.3%

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

       2. unsub-neg 85.3%

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

   16. Simplified 85.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.75e+20)
   (/
    1.0
    (* (+ 2.0 (+ beta alpha)) (/ (* (+ beta 2.0) (+ beta 3.0)) (+ 1.0 beta))))
   (*
    (+ 1.0 (/ (- -1.0 alpha) beta))
    (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75e+20) {
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.75d+20) then
        tmp = 1.0d0 / ((2.0d0 + (beta + alpha)) * (((beta + 2.0d0) * (beta + 3.0d0)) / (1.0d0 + beta)))
    else
        tmp = (1.0d0 + (((-1.0d0) - alpha) / beta)) * (((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.75e+20) {
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.75e+20:
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)))
	else:
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.75e+20)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(beta + alpha)) * Float64(Float64(Float64(beta + 2.0) * Float64(beta + 3.0)) / Float64(1.0 + beta))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) * Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.75e+20)
		tmp = 1.0 / ((2.0 + (beta + alpha)) * (((beta + 2.0) * (beta + 3.0)) / (1.0 + beta)));
	else
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.75e+20], N[(1.0 / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.75e20

   1. Initial program 99.9%

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

   3. Simplified 93.4%

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

      1. associate-+r+ 93.4%

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

      2. fma-undefine 93.4%

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

      3. *-commutative 93.4%

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

      4. associate-+l+ 93.4%

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

      5. +-commutative 93.4%

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

      6. associate-+l+ 93.4%

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

      7. *-commutative 93.4%

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

      8. associate-*r* 93.4%

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

      9. associate-+r+ 93.4%

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

      10. +-commutative 93.4%

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

      11. associate-/l/ 99.7%

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

      12. clear-num 99.7%

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

      13. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-undefine 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+r+ 99.7%

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

      13. distribute-rgt1-in 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 64.2%

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

   if 2.75e20 < beta

   1. Initial program 81.5%

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

   3. Simplified 62.2%

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

      1. associate-+r+ 62.2%

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

      2. fma-undefine 62.2%

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

      3. *-commutative 62.2%

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

      4. associate-+l+ 62.2%

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

      5. +-commutative 62.2%

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

      6. associate-+l+ 62.2%

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

      7. *-commutative 62.2%

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

      8. associate-*r* 62.2%

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

      9. associate-+r+ 62.2%

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

      10. +-commutative 62.2%

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

      11. associate-/l/ 78.1%

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

      12. clear-num 78.1%

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

      13. inv-pow 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. unpow-1 78.1%

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

      2. associate-/l* 80.6%

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

      3. associate-+r+ 80.6%

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

      4. +-commutative 80.6%

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

      5. +-commutative 80.6%

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

      6. associate-+r+ 80.6%

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

      7. +-commutative 80.6%

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

      8. fma-undefine 80.6%

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

      9. +-commutative 80.6%

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

      10. *-commutative 80.6%

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

      11. +-commutative 80.6%

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

      12. associate-+r+ 80.6%

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

      13. distribute-rgt1-in 80.6%

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

      14. +-commutative 80.6%

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

      15. associate-+r+ 80.6%

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

      16. +-commutative 80.6%

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

      17. +-commutative 80.6%

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

   8. Simplified 80.6%

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

      1. associate-*r/ 78.1%

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

      2. associate-+l+ 78.1%

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

      3. +-commutative 78.1%

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

      4. clear-num 78.1%

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

      5. *-un-lft-identity 78.1%

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

      6. associate-/l* 89.2%

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

      7. associate-+r+ 89.2%

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

      8. +-commutative 89.2%

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

      9. +-commutative 89.2%

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

      10. associate-+r+ 89.2%

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

      11. +-commutative 89.2%

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

      12. +-commutative 89.2%

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

      13. +-commutative 89.2%

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

   10. Applied egg-rr 89.2%

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

       1. *-lft-identity 89.2%

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

       2. times-frac 99.7%

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

       3. +-commutative 99.7%

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

       4. +-commutative 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in beta around inf 87.0%

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

   14. Taylor expanded in beta around inf 86.4%

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

       1. mul-1-neg 86.4%

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

       2. unsub-neg 86.4%

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

   16. Simplified 86.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{1 + \beta}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 6.8e+15)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) t_0) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 6.8e+15) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 6.8d+15) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 6.8e+15) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 6.8e+15:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 6.8e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 6.8e+15)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.8e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.8e15

   1. Initial program 99.9%

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

   3. Simplified 93.3%

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

      1. associate-+r+ 93.3%

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

      2. fma-undefine 93.3%

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

      3. *-commutative 93.3%

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

      4. associate-+l+ 93.3%

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

      5. +-commutative 93.3%

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

      6. associate-+l+ 93.3%

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

      7. *-commutative 93.3%

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

      8. associate-*r* 93.3%

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

      9. associate-+r+ 93.3%

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

      10. +-commutative 93.3%

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

      11. associate-/l/ 99.7%

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

      12. clear-num 99.7%

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

      13. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-undefine 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+r+ 99.7%

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

      13. distribute-rgt1-in 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. associate-/l* 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. times-frac 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around 0 64.4%

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

       1. associate-/r* 64.4%

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

       2. +-commutative 64.4%

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

   15. Simplified 64.4%

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

       1. *-commutative 64.4%

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

       2. associate-/l/ 64.4%

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

       3. frac-times 64.4%

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

       4. *-un-lft-identity 64.4%

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

       5. +-commutative 64.4%

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

       6. associate-+r+ 64.4%

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

   17. Applied egg-rr 64.4%

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

   if 6.8e15 < beta

   1. Initial program 81.8%

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

   3. Simplified 62.6%

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

   6. Applied egg-rr 78.4%

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

      1. associate-*r/ 78.4%

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

      2. *-rgt-identity 78.4%

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

      3. fma-undefine 78.4%

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

      4. +-commutative 78.4%

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

      5. *-commutative 78.4%

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

      6. +-commutative 78.4%

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

      7. associate-+r+ 78.4%

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

      8. distribute-rgt1-in 78.4%

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

      9. +-commutative 78.4%

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

      10. associate-+r+ 78.4%

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

      11. +-commutative 78.4%

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

      12. +-commutative 78.4%

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

      13. associate-+r+ 78.4%

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

      14. +-commutative 78.4%

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

      15. +-commutative 78.4%

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

      16. +-commutative 78.4%

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

      17. +-commutative 78.4%

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

      18. +-commutative 78.4%

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

      19. associate-+l+ 78.4%

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

   8. Simplified 78.4%

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

   9. Taylor expanded in beta around inf 87.2%

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

       1. *-un-lft-identity 87.2%

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

       2. associate-/r* 86.1%

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

   11. Applied egg-rr 86.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \beta \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.1)
   (*
    (/ (+ 1.0 beta) (+ alpha (+ beta 2.0)))
    (+ 0.16666666666666666 (* beta -0.1388888888888889)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.1d0) then
        tmp = ((1.0d0 + beta) / (alpha + (beta + 2.0d0))) * (0.16666666666666666d0 + (beta * (-0.1388888888888889d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.1:
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * -0.1388888888888889))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.1)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(alpha + Float64(beta + 2.0))) * Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.1)
		tmp = ((1.0 + beta) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (beta * -0.1388888888888889));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.1], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1000000000000001

   1. Initial program 99.9%

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

   3. Simplified 93.2%

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

      1. associate-+r+ 93.2%

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

      2. fma-undefine 93.2%

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

      3. *-commutative 93.2%

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

      4. associate-+l+ 93.2%

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

      5. +-commutative 93.2%

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

      6. associate-+l+ 93.2%

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

      7. *-commutative 93.2%

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

      8. associate-*r* 93.2%

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

      9. associate-+r+ 93.2%

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

      10. +-commutative 93.2%

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

      11. associate-/l/ 99.7%

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

      12. clear-num 99.7%

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

      13. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-undefine 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+r+ 99.7%

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

      13. distribute-rgt1-in 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. associate-/l* 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. times-frac 99.7%

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

       3. +-commutative 99.7%

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

       4. +-commutative 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in alpha around 0 64.7%

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

       1. associate-/r* 64.7%

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

       2. +-commutative 64.7%

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

   15. Simplified 64.7%

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

   16. Taylor expanded in beta around 0 64.4%

       \[\leadsto \frac{1 + \beta}{\alpha + \left(2 + \beta\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \beta\right)} \]

   if 1.1000000000000001 < beta

   1. Initial program 82.6%

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

   3. Taylor expanded in beta around inf 83.2%

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

   4. Taylor expanded in alpha around 0 83.2%

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

      1. associate-+r+ 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

   6. Simplified 83.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \beta \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 29:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 29.0)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 3.0) (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 29.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 29.0d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 3.0d0) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 29.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 29.0:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 29.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 29.0)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 29.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 29

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-+l+ 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around 0 98.5%

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

   6. Taylor expanded in beta around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. +-commutative 98.6%

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

   8. Simplified 98.6%

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

   if 29 < beta

   1. Initial program 82.6%

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

   3. Taylor expanded in beta around inf 83.2%

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

   4. Taylor expanded in alpha around 0 83.2%

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

      1. associate-+r+ 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

   6. Simplified 83.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 29:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 3.0) (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.55d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 3.0d0) * (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 3.0) * (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.55], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-+l+ 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around 0 98.5%

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

   6. Taylor expanded in beta around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. +-commutative 98.6%

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

   8. Simplified 98.6%

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

   if 1.55000000000000004 < beta

   1. Initial program 82.6%

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

   3. Simplified 64.4%

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

   6. Applied egg-rr 79.3%

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

      1. associate-*r/ 79.4%

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

      2. *-rgt-identity 79.4%

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

      3. fma-undefine 79.4%

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

      4. +-commutative 79.4%

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

      5. *-commutative 79.4%

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

      6. +-commutative 79.4%

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

      7. associate-+r+ 79.4%

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

      8. distribute-rgt1-in 79.4%

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

      9. +-commutative 79.4%

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

      10. associate-+r+ 79.4%

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

      11. +-commutative 79.4%

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

      12. +-commutative 79.4%

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

      13. associate-+r+ 79.4%

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

      14. +-commutative 79.4%

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

      15. +-commutative 79.4%

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

      16. +-commutative 79.4%

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

      17. +-commutative 79.4%

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

      18. +-commutative 79.4%

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

      19. associate-+l+ 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in beta around inf 86.9%

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

       1. *-un-lft-identity 86.9%

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

       2. associate-/r* 83.7%

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

   11. Applied egg-rr 83.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{6 + \alpha \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (+ 6.0 (* alpha 5.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / (6.0 + (alpha * 5.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / (6.0d0 + (alpha * 5.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / (6.0 + (alpha * 5.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / (6.0 + (alpha * 5.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(6.0 + Float64(alpha * 5.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / (6.0 + (alpha * 5.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.4], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(alpha * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-+l+ 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around 0 98.5%

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

   6. Taylor expanded in beta around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. +-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in alpha around 0 64.7%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{6 + 5 \cdot \alpha}} \]
   10. Step-by-step derivation

       1. *-commutative 64.7%

          \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{6 + \color{blue}{\alpha \cdot 5}} \]

   11. Simplified 64.7%

       \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{6 + \alpha \cdot 5}} \]

   if 2.39999999999999991 < beta

   1. Initial program 82.6%

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

   3. Taylor expanded in beta around inf 83.2%

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

   4. Taylor expanded in alpha around 0 83.2%

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

      1. associate-+r+ 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

   6. Simplified 83.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{6 + \alpha \cdot 5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-+l+ 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 98.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   6. Taylor expanded in beta around 0 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

      2. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   8. Simplified 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 2.10000000000000009 < beta

   1. Initial program 82.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 83.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in alpha around 0 83.2%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \left(\alpha + \beta\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 83.2%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\left(3 + \alpha\right) + \beta}} \]

      2. +-commutative 83.2%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\left(\alpha + 3\right)} + \beta} \]

      3. +-commutative 83.2%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta + \left(\alpha + 3\right)}} \]

   6. Simplified 83.2%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta + \left(\alpha + 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.9)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.9d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.9:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.9)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.9)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.9], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8999999999999999

   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/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. metadata-eval 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

      11. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 98.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   6. Taylor expanded in beta around 0 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

      2. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   8. Simplified 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 1.8999999999999999 < beta

   1. Initial program 82.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 83.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in alpha around 0 75.2%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 75.2%

         \[\leadsto \frac{1}{\beta \cdot \color{blue}{\left(\beta + 3\right)}} \]

   6. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(\beta + 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.45)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.45) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.45d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.45) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.45:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.45)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.45)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.45], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4500000000000002

   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/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. metadata-eval 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

      11. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 98.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   6. Taylor expanded in beta around 0 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

      2. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   8. Simplified 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 3.4500000000000002 < beta

   1. Initial program 82.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 83.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 83.2%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. metadata-eval 83.2%

         \[\leadsto 1 \cdot \frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]

      3. associate-+l+ 83.2%

         \[\leadsto 1 \cdot \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]

      4. metadata-eval 83.2%

         \[\leadsto 1 \cdot \frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      5. associate-/l/ 83.4%

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \beta}} \]

      6. +-commutative 83.4%

         \[\leadsto 1 \cdot \frac{1 + \alpha}{\left(\color{blue}{\left(\beta + \alpha\right)} + 3\right) \cdot \beta} \]

      7. associate-+l+ 83.4%

         \[\leadsto 1 \cdot \frac{1 + \alpha}{\color{blue}{\left(\beta + \left(\alpha + 3\right)\right)} \cdot \beta} \]

   5. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{1 \cdot \frac{1 + \alpha}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \beta}} \]
   6. Step-by-step derivation

      1. *-lft-identity 83.4%

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \beta}} \]

      2. *-commutative 83.4%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   7. Simplified 83.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   8. Taylor expanded in beta around inf 81.0%

      \[\leadsto \frac{1 + \alpha}{\beta \cdot \color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.3d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.3:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	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 <= 3.3)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. metadata-eval 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

      11. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 98.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   6. Taylor expanded in beta around 0 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

      2. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   8. Simplified 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 3.2999999999999998 < beta

   1. Initial program 82.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 83.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in beta around inf 82.9%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10.6:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10.6)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.6) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} 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 <= 10.6d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    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 <= 10.6) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 10.6:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10.6)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	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 <= 10.6)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	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, 10.6], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 10.5999999999999996

   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/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. metadata-eval 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

      11. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 98.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   6. Taylor expanded in beta around 0 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

      2. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   8. Simplified 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 10.5999999999999996 < beta

   1. Initial program 82.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 83.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in alpha around inf 6.9%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10.6:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.1% accurate, 4.4× 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/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. metadata-eval 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

      11. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 98.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   6. Taylor expanded in beta around 0 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

      2. +-commutative 98.6%

         \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   8. Simplified 98.6%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

   9. Taylor expanded in alpha around 0 63.1%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 12 < beta

   1. Initial program 82.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 83.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in alpha around inf 6.9%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 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 94.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 92.9%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative 92.9%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. associate-+l+ 92.9%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. *-commutative 92.9%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. metadata-eval 92.9%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. associate-+l+ 92.9%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. metadata-eval 92.9%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. associate-+l+ 92.8%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. metadata-eval 92.8%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. metadata-eval 92.8%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

   11. associate-+l+ 92.8%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

3. Simplified 92.8%

   \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in beta around 0 84.9%

   \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

6. Taylor expanded in beta around 0 70.2%

   \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
7. Step-by-step derivation

   1. +-commutative 70.2%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right)} \cdot \left(3 + \alpha\right)} \]

   2. +-commutative 70.2%

      \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

8. Simplified 70.2%

   \[\leadsto \frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}} \]

9. Taylor expanded in alpha around 0 43.2%

   \[\leadsto \color{blue}{0.08333333333333333} \]

10. Final simplification 43.2%

    \[\leadsto 0.08333333333333333 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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)))