Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 21.2s

Alternatives: 16

Speedup: 2.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

   1. associate-/l/ 94.0%

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

   2. +-commutative 94.0%

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

   3. associate-+l+ 94.0%

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

   4. *-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. metadata-eval 94.0%

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

   7. associate-+l+ 94.0%

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

   8. metadata-eval 94.0%

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

   9. associate-+l+ 94.0%

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

   10. +-commutative 94.0%

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

   11. associate-+l+ 94.0%

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

   12. metadata-eval 94.0%

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

   13. +-commutative 94.0%

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

   14. metadata-eval 94.0%

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

3. Simplified 94.0%

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

   1. clear-num 94.0%

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

   2. inv-pow 94.0%

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

   3. associate-+r+ 94.0%

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

   4. associate-+r+ 94.0%

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

   5. +-commutative 94.0%

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

   6. associate-+r+ 94.0%

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

   7. +-commutative 94.0%

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

   8. distribute-rgt1-in 94.0%

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

   9. fma-define 94.0%

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

   10. associate-+r+ 94.0%

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

   11. +-commutative 94.0%

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

   12. associate-+r+ 94.0%

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

6. Applied egg-rr 94.0%

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

   1. unpow-1 94.0%

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

   2. associate-/r/ 94.0%

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

   3. *-commutative 94.0%

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

   4. +-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. +-commutative 94.0%

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

   7. fma-undefine 94.0%

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

   8. +-commutative 94.0%

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

   9. *-commutative 94.0%

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

   10. +-commutative 94.0%

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

   11. associate-+r+ 94.0%

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

   12. distribute-lft1-in 94.0%

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

   13. +-commutative 94.0%

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

   14. +-commutative 94.0%

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

   15. *-commutative 94.0%

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

   16. +-commutative 94.0%

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

8. Simplified 94.0%

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

   1. inv-pow 94.0%

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

   2. *-commutative 94.0%

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

   3. +-commutative 94.0%

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

   4. +-commutative 94.0%

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

   5. unpow-prod-down 94.0%

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

   6. inv-pow 94.0%

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

   7. inv-pow 94.0%

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

   8. times-frac 99.7%

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

   9. +-commutative 99.7%

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

   10. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

12. Simplified 99.8%

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

    1. add0 99.8%

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

    2. *-rgt-identity 99.8%

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

14. Applied egg-rr 99.8%

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

    1. add0 99.8%

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

    2. associate-+r+ 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-+r+ 99.8%

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

    5. +-commutative 99.8%

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

    6. associate-+r+ 99.8%

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

    7. +-commutative 99.8%

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

    8. associate-+r+ 99.8%

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

    9. +-commutative 99.8%

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

    10. +-commutative 99.8%

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

    11. +-commutative 99.8%

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

    12. +-commutative 99.8%

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

16. Simplified 99.8%

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

17. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 + \beta}{t\_1} \cdot \frac{1 + \alpha}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{2 + \left(\beta + \alpha\right)}{1 + \alpha} \cdot \frac{t\_1}{1 + \beta}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))) (t_1 (+ alpha (+ beta 3.0))))
   (if (<= beta 3.8e+133)
     (* (/ (+ 1.0 beta) t_1) (/ (+ 1.0 alpha) (* t_0 t_0)))
     (/
      (/ 1.0 beta)
      (* (/ (+ 2.0 (+ beta alpha)) (+ 1.0 alpha)) (/ t_1 (+ 1.0 beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.8e+133) {
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0));
	} else {
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * (t_1 / (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    t_1 = alpha + (beta + 3.0d0)
    if (beta <= 3.8d+133) then
        tmp = ((1.0d0 + beta) / t_1) * ((1.0d0 + alpha) / (t_0 * t_0))
    else
        tmp = (1.0d0 / beta) / (((2.0d0 + (beta + alpha)) / (1.0d0 + alpha)) * (t_1 / (1.0d0 + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.8e+133) {
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0));
	} else {
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * (t_1 / (1.0 + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	t_1 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 3.8e+133:
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0))
	else:
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * (t_1 / (1.0 + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	t_1 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 3.8e+133)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_1) * Float64(Float64(1.0 + alpha) / Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(Float64(Float64(2.0 + Float64(beta + alpha)) / Float64(1.0 + alpha)) * Float64(t_1 / Float64(1.0 + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	t_1 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 3.8e+133)
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0));
	else
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * (t_1 / (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_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.8e+133], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8000000000000002e133

   1. Initial program 98.4%

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

   3. Simplified 98.3%

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

   if 3.8000000000000002e133 < beta

   1. Initial program 83.5%

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

      1. associate-/l/ 79.3%

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

      2. +-commutative 79.3%

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

      3. associate-+l+ 79.3%

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

      4. *-commutative 79.3%

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

      5. +-commutative 79.3%

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

      6. metadata-eval 79.3%

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

      7. associate-+l+ 79.3%

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

      8. metadata-eval 79.3%

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

      9. associate-+l+ 79.3%

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

      10. +-commutative 79.3%

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

      11. associate-+l+ 79.3%

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

      12. metadata-eval 79.3%

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

      13. +-commutative 79.3%

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

      14. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

      1. clear-num 79.3%

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

      2. inv-pow 79.3%

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

      3. associate-+r+ 79.3%

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

      4. associate-+r+ 79.3%

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

      5. +-commutative 79.3%

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

      6. associate-+r+ 79.3%

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

      7. +-commutative 79.3%

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

      8. distribute-rgt1-in 79.3%

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

      9. fma-define 79.3%

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

      10. associate-+r+ 79.3%

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

      11. +-commutative 79.3%

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

      12. associate-+r+ 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. unpow-1 79.3%

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

      2. associate-/r/ 79.4%

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

      3. *-commutative 79.4%

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

      4. +-commutative 79.4%

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

      5. +-commutative 79.4%

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

      6. +-commutative 79.4%

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

      7. fma-undefine 79.4%

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

      8. +-commutative 79.4%

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

      9. *-commutative 79.4%

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

      10. +-commutative 79.4%

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

      11. associate-+r+ 79.4%

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

      12. distribute-lft1-in 79.4%

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

      13. +-commutative 79.4%

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

      14. +-commutative 79.4%

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

      15. *-commutative 79.4%

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

      16. +-commutative 79.4%

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

   8. Simplified 79.4%

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

      1. inv-pow 79.4%

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

      2. *-commutative 79.4%

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

      3. +-commutative 79.4%

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

      4. +-commutative 79.4%

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

      5. unpow-prod-down 79.4%

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

      6. inv-pow 79.4%

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

      7. inv-pow 79.4%

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

      8. times-frac 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

   12. Simplified 99.8%

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

       1. add0 99.8%

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

       2. *-rgt-identity 99.8%

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

   14. Applied egg-rr 99.8%

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

       1. add0 99.8%

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

       2. associate-+r+ 99.8%

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

       3. +-commutative 99.8%

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

       4. associate-+r+ 99.8%

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

       5. +-commutative 99.8%

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

       6. associate-+r+ 99.8%

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

       7. +-commutative 99.8%

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

       8. associate-+r+ 99.8%

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

       9. +-commutative 99.8%

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

       10. +-commutative 99.8%

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

       11. +-commutative 99.8%

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

       12. +-commutative 99.8%

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

   16. Simplified 99.8%

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

   17. Taylor expanded in beta around inf 92.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{2 + \left(\beta + \alpha\right)}{1 + \alpha} \cdot \frac{\alpha + \left(\beta + 3\right)}{1 + \beta}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 + \beta}{t\_1} \cdot \frac{1 + \alpha}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{\frac{t\_1}{1 + \beta} \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))) (t_1 (+ alpha (+ beta 3.0))))
   (if (<= beta 3.8e+133)
     (* (/ (+ 1.0 beta) t_1) (/ (+ 1.0 alpha) (* t_0 t_0)))
     (/ (/ 1.0 t_0) (* (/ t_1 (+ 1.0 beta)) (/ beta (+ 1.0 alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.8e+133) {
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0));
	} else {
		tmp = (1.0 / t_0) / ((t_1 / (1.0 + beta)) * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    t_1 = alpha + (beta + 3.0d0)
    if (beta <= 3.8d+133) then
        tmp = ((1.0d0 + beta) / t_1) * ((1.0d0 + alpha) / (t_0 * t_0))
    else
        tmp = (1.0d0 / t_0) / ((t_1 / (1.0d0 + beta)) * (beta / (1.0d0 + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.8e+133) {
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0));
	} else {
		tmp = (1.0 / t_0) / ((t_1 / (1.0 + beta)) * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	t_1 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 3.8e+133:
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0))
	else:
		tmp = (1.0 / t_0) / ((t_1 / (1.0 + beta)) * (beta / (1.0 + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	t_1 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 3.8e+133)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_1) * Float64(Float64(1.0 + alpha) / Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(1.0 / t_0) / Float64(Float64(t_1 / Float64(1.0 + beta)) * Float64(beta / Float64(1.0 + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	t_1 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 3.8e+133)
		tmp = ((1.0 + beta) / t_1) * ((1.0 + alpha) / (t_0 * t_0));
	else
		tmp = (1.0 / t_0) / ((t_1 / (1.0 + beta)) * (beta / (1.0 + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.8e+133], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(N[(t$95$1 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8000000000000002e133

   1. Initial program 98.4%

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

   3. Simplified 98.3%

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

   if 3.8000000000000002e133 < beta

   1. Initial program 83.5%

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

      1. associate-/l/ 79.3%

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

      2. +-commutative 79.3%

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

      3. associate-+l+ 79.3%

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

      4. *-commutative 79.3%

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

      5. +-commutative 79.3%

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

      6. metadata-eval 79.3%

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

      7. associate-+l+ 79.3%

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

      8. metadata-eval 79.3%

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

      9. associate-+l+ 79.3%

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

      10. +-commutative 79.3%

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

      11. associate-+l+ 79.3%

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

      12. metadata-eval 79.3%

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

      13. +-commutative 79.3%

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

      14. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

      1. clear-num 79.3%

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

      2. inv-pow 79.3%

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

      3. associate-+r+ 79.3%

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

      4. associate-+r+ 79.3%

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

      5. +-commutative 79.3%

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

      6. associate-+r+ 79.3%

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

      7. +-commutative 79.3%

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

      8. distribute-rgt1-in 79.3%

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

      9. fma-define 79.3%

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

      10. associate-+r+ 79.3%

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

      11. +-commutative 79.3%

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

      12. associate-+r+ 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. unpow-1 79.3%

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

      2. associate-/r/ 79.4%

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

      3. *-commutative 79.4%

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

      4. +-commutative 79.4%

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

      5. +-commutative 79.4%

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

      6. +-commutative 79.4%

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

      7. fma-undefine 79.4%

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

      8. +-commutative 79.4%

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

      9. *-commutative 79.4%

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

      10. +-commutative 79.4%

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

      11. associate-+r+ 79.4%

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

      12. distribute-lft1-in 79.4%

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

      13. +-commutative 79.4%

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

      14. +-commutative 79.4%

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

      15. *-commutative 79.4%

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

      16. +-commutative 79.4%

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

   8. Simplified 79.4%

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

      1. inv-pow 79.4%

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

      2. *-commutative 79.4%

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

      3. +-commutative 79.4%

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

      4. +-commutative 79.4%

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

      5. unpow-prod-down 79.4%

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

      6. inv-pow 79.4%

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

      7. inv-pow 79.4%

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

      8. times-frac 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in beta around inf 92.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(\beta + 3\right)}{1 + \beta} \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+36}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{2 + \left(\beta + \alpha\right)}{1 + \alpha} \cdot \frac{\alpha + \left(\beta + 3\right)}{1 + \beta}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+36)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ 3.0 (+ beta alpha))))
   (/
    (/ 1.0 beta)
    (*
     (/ (+ 2.0 (+ beta alpha)) (+ 1.0 alpha))
     (/ (+ alpha (+ beta 3.0)) (+ 1.0 beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+36) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)));
	} else {
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * ((alpha + (beta + 3.0)) / (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 <= 1d+36) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (3.0d0 + (beta + alpha)))
    else
        tmp = (1.0d0 / beta) / (((2.0d0 + (beta + alpha)) / (1.0d0 + alpha)) * ((alpha + (beta + 3.0d0)) / (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 <= 1e+36) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)));
	} else {
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * ((alpha + (beta + 3.0)) / (1.0 + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1e+36:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)))
	else:
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * ((alpha + (beta + 3.0)) / (1.0 + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+36)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(Float64(Float64(2.0 + Float64(beta + alpha)) / Float64(1.0 + alpha)) * Float64(Float64(alpha + Float64(beta + 3.0)) / 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 <= 1e+36)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)));
	else
		tmp = (1.0 / beta) / (((2.0 + (beta + alpha)) / (1.0 + alpha)) * ((alpha + (beta + 3.0)) / (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, 1e+36], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000004e36

   1. Initial program 99.7%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.2%

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

   6. Taylor expanded in alpha around 0 66.3%

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

   if 1.00000000000000004e36 < beta

   1. Initial program 86.7%

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

      1. associate-/l/ 80.1%

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

      2. +-commutative 80.1%

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

      3. associate-+l+ 80.1%

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

      4. *-commutative 80.1%

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

      5. +-commutative 80.1%

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

      6. metadata-eval 80.1%

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

      7. associate-+l+ 80.1%

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

      8. metadata-eval 80.1%

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

      9. associate-+l+ 80.1%

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

      10. +-commutative 80.1%

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

      11. associate-+l+ 80.1%

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

      12. metadata-eval 80.1%

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

      13. +-commutative 80.1%

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

      14. metadata-eval 80.1%

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

   3. Simplified 80.1%

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

      1. clear-num 80.1%

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

      2. inv-pow 80.1%

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

      3. associate-+r+ 80.1%

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

      4. associate-+r+ 80.1%

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

      5. +-commutative 80.1%

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

      6. associate-+r+ 80.1%

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

      7. +-commutative 80.1%

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

      8. distribute-rgt1-in 80.1%

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

      9. fma-define 80.1%

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

      10. associate-+r+ 80.1%

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

      11. +-commutative 80.1%

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

      12. associate-+r+ 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. unpow-1 80.1%

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

      2. associate-/r/ 80.1%

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

      3. *-commutative 80.1%

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

      4. +-commutative 80.1%

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

      5. +-commutative 80.1%

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

      6. +-commutative 80.1%

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

      7. fma-undefine 80.1%

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

      8. +-commutative 80.1%

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

      9. *-commutative 80.1%

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

      10. +-commutative 80.1%

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

      11. associate-+r+ 80.1%

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

      12. distribute-lft1-in 80.1%

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

      13. +-commutative 80.1%

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

      14. +-commutative 80.1%

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

      15. *-commutative 80.1%

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

      16. +-commutative 80.1%

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

   8. Simplified 80.1%

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

      1. inv-pow 80.1%

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

      2. *-commutative 80.1%

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

      3. +-commutative 80.1%

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

      4. +-commutative 80.1%

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

      5. unpow-prod-down 80.1%

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

      6. inv-pow 80.1%

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

      7. inv-pow 80.1%

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

      8. times-frac 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*r/ 99.8%

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

       2. +-commutative 99.8%

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

       3. +-commutative 99.8%

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

   12. Simplified 99.8%

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

       1. add0 99.8%

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

       2. *-rgt-identity 99.8%

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

   14. Applied egg-rr 99.8%

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

       1. add0 99.8%

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

       2. associate-+r+ 99.8%

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

       3. +-commutative 99.8%

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

       4. associate-+r+ 99.8%

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

       5. +-commutative 99.8%

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

       6. associate-+r+ 99.8%

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

       7. +-commutative 99.8%

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

       8. associate-+r+ 99.8%

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

       9. +-commutative 99.8%

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

       10. +-commutative 99.8%

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

       11. +-commutative 99.8%

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

       12. +-commutative 99.8%

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

   16. Simplified 99.8%

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

   17. Taylor expanded in beta around inf 84.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+36}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{2 + \left(\beta + \alpha\right)}{1 + \alpha} \cdot \frac{\alpha + \left(\beta + 3\right)}{1 + \beta}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

   1. associate-/l/ 94.0%

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

   2. +-commutative 94.0%

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

   3. associate-+l+ 94.0%

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

   4. *-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. metadata-eval 94.0%

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

   7. associate-+l+ 94.0%

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

   8. metadata-eval 94.0%

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

   9. associate-+l+ 94.0%

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

   10. +-commutative 94.0%

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

   11. associate-+l+ 94.0%

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

   12. metadata-eval 94.0%

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

   13. +-commutative 94.0%

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

   14. metadata-eval 94.0%

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

3. Simplified 94.0%

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

   1. clear-num 94.0%

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

   2. inv-pow 94.0%

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

   3. associate-+r+ 94.0%

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

   4. associate-+r+ 94.0%

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

   5. +-commutative 94.0%

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

   6. associate-+r+ 94.0%

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

   7. +-commutative 94.0%

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

   8. distribute-rgt1-in 94.0%

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

   9. fma-define 94.0%

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

   10. associate-+r+ 94.0%

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

   11. +-commutative 94.0%

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

   12. associate-+r+ 94.0%

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

6. Applied egg-rr 94.0%

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

   1. unpow-1 94.0%

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

   2. associate-/r/ 94.0%

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

   3. *-commutative 94.0%

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

   4. +-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. +-commutative 94.0%

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

   7. fma-undefine 94.0%

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

   8. +-commutative 94.0%

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

   9. *-commutative 94.0%

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

   10. +-commutative 94.0%

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

   11. associate-+r+ 94.0%

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

   12. distribute-lft1-in 94.0%

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

   13. +-commutative 94.0%

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

   14. +-commutative 94.0%

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

   15. *-commutative 94.0%

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

   16. +-commutative 94.0%

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

8. Simplified 94.0%

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

   1. times-frac 99.3%

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

   2. +-commutative 99.3%

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

   3. +-commutative 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 6: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{-50}:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.65e-50)
   (/ (+ 1.0 alpha) (* (+ 2.0 alpha) (* (+ 2.0 alpha) (+ alpha 3.0))))
   (if (<= beta 1.45e+36)
     (/ (+ 1.0 beta) (* (+ 2.0 beta) (* (+ beta 3.0) (+ 2.0 beta))))
     (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65e-50) {
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)));
	} else if (beta <= 1.45e+36) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.65d-50) then
        tmp = (1.0d0 + alpha) / ((2.0d0 + alpha) * ((2.0d0 + alpha) * (alpha + 3.0d0)))
    else if (beta <= 1.45d+36) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65e-50) {
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)));
	} else if (beta <= 1.45e+36) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.65e-50:
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)))
	elif beta <= 1.45e+36:
		tmp = (1.0 + beta) / ((2.0 + beta) * ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.65e-50)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + alpha) * Float64(Float64(2.0 + alpha) * Float64(alpha + 3.0))));
	elseif (beta <= 1.45e+36)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.65e-50)
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)));
	elseif (beta <= 1.45e+36)
		tmp = (1.0 + beta) / ((2.0 + beta) * ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.65e-50], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.45e+36], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.6499999999999999e-50

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-+r+ 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+r+ 99.8%

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

      8. distribute-lft-in 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in beta around 0 90.3%

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

      1. +-commutative 90.3%

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

      2. distribute-rgt-in 90.2%

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

      3. +-commutative 90.2%

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

      4. +-commutative 90.2%

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

   9. Simplified 90.2%

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

   if 1.6499999999999999e-50 < beta < 1.45e36

   1. Initial program 99.3%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. metadata-eval 99.6%

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

      7. associate-+l+ 99.6%

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

      8. metadata-eval 99.6%

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

      9. associate-+l+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+l+ 99.6%

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

      12. metadata-eval 99.6%

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

      13. +-commutative 99.6%

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

      14. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 90.4%

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

      1. add0 90.4%

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

      2. associate-/l/ 90.4%

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

      3. associate-+r+ 90.4%

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

      4. *-commutative 90.4%

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

      5. associate-+r+ 90.4%

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

      6. +-commutative 90.4%

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

      7. associate-+r+ 90.4%

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

      8. +-commutative 90.4%

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

   7. Applied egg-rr 90.4%

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

      1. add0 90.4%

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

      2. +-commutative 90.4%

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

      3. *-commutative 90.4%

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

      4. +-commutative 90.4%

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

   9. Simplified 90.4%

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

   10. Taylor expanded in alpha around 0 71.0%

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

       1. +-commutative 71.0%

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

       2. +-commutative 71.0%

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

   12. Simplified 71.0%

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

   if 1.45e36 < beta

   1. Initial program 86.7%

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

   3. Simplified 89.6%

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

      1. associate-*l/ 89.6%

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

      2. +-commutative 89.6%

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

      3. div-inv 89.6%

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

      4. pow2 89.6%

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

      5. associate-+r+ 89.6%

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

      6. metadata-eval 89.6%

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

      7. pow-flip 90.7%

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

      8. metadata-eval 90.7%

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

      9. associate-+r+ 90.7%

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

      10. metadata-eval 90.7%

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

   6. Applied egg-rr 90.7%

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

   7. Taylor expanded in beta around inf 84.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{-50}:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.5e+35)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ 3.0 (+ beta alpha))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5e+35) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.5d+35) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (3.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5e+35) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.5e+35:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5e+35)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.5e+35)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (3.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.5e+35], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.50000000000000001e35

   1. Initial program 99.7%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.2%

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

   6. Taylor expanded in alpha around 0 66.3%

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

   if 5.50000000000000001e35 < beta

   1. Initial program 86.7%

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

   3. Simplified 89.6%

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

      1. associate-*l/ 89.6%

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

      2. +-commutative 89.6%

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

      3. div-inv 89.6%

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

      4. pow2 89.6%

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

      5. associate-+r+ 89.6%

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

      6. metadata-eval 89.6%

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

      7. pow-flip 90.7%

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

      8. metadata-eval 90.7%

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

      9. associate-+r+ 90.7%

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

      10. metadata-eval 90.7%

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

   6. Applied egg-rr 90.7%

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

   7. Taylor expanded in beta around inf 84.1%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 2.5)
     (/ 0.25 (+ alpha 3.0))
     (if (<= beta 3.8e+154)
       (/ (+ 1.0 alpha) (* beta t_0))
       (/ (/ alpha beta) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.25 / (alpha + 3.0);
	} else if (beta <= 3.8e+154) {
		tmp = (1.0 + alpha) / (beta * t_0);
	} else {
		tmp = (alpha / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (beta <= 2.5d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else if (beta <= 3.8d+154) then
        tmp = (1.0d0 + alpha) / (beta * t_0)
    else
        tmp = (alpha / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.25 / (alpha + 3.0);
	} else if (beta <= 3.8e+154) {
		tmp = (1.0 + alpha) / (beta * t_0);
	} else {
		tmp = (alpha / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 2.5:
		tmp = 0.25 / (alpha + 3.0)
	elif beta <= 3.8e+154:
		tmp = (1.0 + alpha) / (beta * t_0)
	else:
		tmp = (alpha / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	elseif (beta <= 3.8e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * t_0));
	else
		tmp = Float64(Float64(alpha / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.25 / (alpha + 3.0);
	elseif (beta <= 3.8e+154)
		tmp = (1.0 + alpha) / (beta * t_0);
	else
		tmp = (alpha / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.8e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.0%

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

   6. Taylor expanded in alpha around 0 65.3%

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

   7. Taylor expanded in beta around 0 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]
   8. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{\alpha + 3}} \]

   if 2.5 < beta < 3.7999999999999998e154

   1. Initial program 93.0%

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

   3. Simplified 92.6%

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

      1. associate-*l/ 92.4%

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

      2. +-commutative 92.4%

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

      3. div-inv 92.4%

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

      4. pow2 92.4%

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

      5. associate-+r+ 92.4%

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

      6. metadata-eval 92.4%

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

      7. pow-flip 92.6%

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

      8. metadata-eval 92.6%

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

      9. associate-+r+ 92.6%

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

      10. metadata-eval 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in beta around inf 75.7%

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

      1. add0 75.7%

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

      2. associate-/l/ 82.2%

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

   9. Applied egg-rr 82.2%

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

       1. add0 82.2%

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

       2. *-commutative 82.2%

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

       3. +-commutative 82.2%

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

       4. +-commutative 82.2%

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

       5. +-commutative 82.2%

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

   11. Simplified 82.2%

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

   if 3.7999999999999998e154 < beta

   1. Initial program 81.9%

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

   3. Simplified 88.3%

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

      1. associate-*l/ 88.3%

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

      2. +-commutative 88.3%

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

      3. div-inv 88.3%

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

      4. pow2 88.3%

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

      5. associate-+r+ 88.3%

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

      6. metadata-eval 88.3%

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

      7. pow-flip 90.1%

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

      8. metadata-eval 90.1%

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

      9. associate-+r+ 90.1%

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

      10. metadata-eval 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in beta around inf 91.5%

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

   8. Taylor expanded in alpha around inf 89.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (/ (+ 1.0 alpha) (* (+ 2.0 alpha) (* (+ 2.0 alpha) (+ alpha 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = (1.0d0 + alpha) / ((2.0d0 + alpha) * ((2.0d0 + alpha) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + alpha) * Float64(Float64(2.0 + alpha) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = (1.0 + alpha) / ((2.0 + alpha) * ((2.0 + alpha) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-+r+ 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+r+ 99.8%

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

      8. distribute-lft-in 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around 0 90.3%

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

      1. +-commutative 90.3%

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

      2. distribute-rgt-in 90.3%

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

      3. +-commutative 90.3%

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

      4. +-commutative 90.3%

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

   9. Simplified 90.3%

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

   if 2.60000000000000009 < beta

   1. Initial program 87.9%

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

   3. Simplified 90.6%

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

      1. associate-*l/ 90.5%

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

      2. +-commutative 90.5%

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

      3. div-inv 90.5%

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

      4. pow2 90.5%

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

      5. associate-+r+ 90.5%

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

      6. metadata-eval 90.5%

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

      7. pow-flip 91.5%

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

      8. metadata-eval 91.5%

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

      9. associate-+r+ 91.5%

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

      10. metadata-eval 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (/ 0.25 (+ alpha 3.0))
   (if (<= beta 8e+161)
     (/ (/ 1.0 (+ beta 3.0)) (+ 2.0 beta))
     (/ (/ alpha beta) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.25 / (alpha + 3.0);
	} else if (beta <= 8e+161) {
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta);
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else if (beta <= 8d+161) then
        tmp = (1.0d0 / (beta + 3.0d0)) / (2.0d0 + beta)
    else
        tmp = (alpha / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.25 / (alpha + 3.0);
	} else if (beta <= 8e+161) {
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta);
	} else {
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.25 / (alpha + 3.0)
	elif beta <= 8e+161:
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta)
	else:
		tmp = (alpha / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	elseif (beta <= 8e+161)
		tmp = Float64(Float64(1.0 / Float64(beta + 3.0)) / Float64(2.0 + beta));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.25 / (alpha + 3.0);
	elseif (beta <= 8e+161)
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta);
	else
		tmp = (alpha / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.2], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8e+161], N[(N[(1.0 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.19999999999999996

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.0%

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

   6. Taylor expanded in alpha around 0 65.3%

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

   7. Taylor expanded in beta around 0 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]
   8. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{\alpha + 3}} \]

   if 1.19999999999999996 < beta < 8.0000000000000003e161

   1. Initial program 91.4%

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

      1. associate-/l/ 83.3%

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

      2. +-commutative 83.3%

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

      3. associate-+l+ 83.3%

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

      4. *-commutative 83.3%

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

      5. +-commutative 83.3%

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

      6. metadata-eval 83.3%

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

      7. associate-+l+ 83.3%

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

      8. metadata-eval 83.3%

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

      9. associate-+l+ 83.3%

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

      10. +-commutative 83.3%

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

      11. associate-+l+ 83.3%

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

      12. metadata-eval 83.3%

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

      13. +-commutative 83.3%

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

      14. metadata-eval 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in alpha around 0 81.8%

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

   6. Taylor expanded in alpha around 0 72.6%

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

      1. +-commutative 72.6%

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

   8. Simplified 72.6%

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

   9. Taylor expanded in beta around inf 70.2%

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

       1. add0 70.2%

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

       2. *-commutative 70.2%

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

   11. Applied egg-rr 70.2%

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

       1. add0 70.2%

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

       2. associate-/r* 71.6%

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

       3. +-commutative 71.6%

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

   13. Simplified 71.6%

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

   if 8.0000000000000003e161 < beta

   1. Initial program 83.1%

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

   3. Simplified 92.0%

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

      1. associate-*l/ 92.0%

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

      2. +-commutative 92.0%

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

      3. div-inv 92.0%

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

      4. pow2 92.0%

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

      5. associate-+r+ 92.0%

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

      6. metadata-eval 92.0%

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

      7. pow-flip 92.0%

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

      8. metadata-eval 92.0%

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

      9. associate-+r+ 92.0%

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

      10. metadata-eval 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in beta around inf 93.4%

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

   8. Taylor expanded in alpha around inf 93.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (/ 0.25 (+ alpha 3.0))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.0%

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

   6. Taylor expanded in alpha around 0 65.3%

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

   7. Taylor expanded in beta around 0 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]
   8. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{\alpha + 3}} \]

   if 2.5 < beta

   1. Initial program 87.9%

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

   3. Simplified 90.6%

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

      1. associate-*l/ 90.5%

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

      2. +-commutative 90.5%

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

      3. div-inv 90.5%

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

      4. pow2 90.5%

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

      5. associate-+r+ 90.5%

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

      6. metadata-eval 90.5%

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

      7. pow-flip 91.5%

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

      8. metadata-eval 91.5%

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

      9. associate-+r+ 91.5%

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

      10. metadata-eval 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{2 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.1)
   (/ 0.25 (+ alpha 3.0))
   (/ (/ 1.0 (+ beta 3.0)) (+ 2.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.1d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else
        tmp = (1.0d0 / (beta + 3.0d0)) / (2.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.1:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.1)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(beta + 3.0)) / Float64(2.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.1)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = (1.0 / (beta + 3.0)) / (2.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.1], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1000000000000001

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.0%

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

   6. Taylor expanded in alpha around 0 65.3%

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

   7. Taylor expanded in beta around 0 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]
   8. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{\alpha + 3}} \]

   if 1.1000000000000001 < beta

   1. Initial program 87.9%

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

      1. associate-/l/ 82.0%

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

      2. +-commutative 82.0%

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

      3. associate-+l+ 82.0%

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

      4. *-commutative 82.0%

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

      5. +-commutative 82.0%

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

      6. metadata-eval 82.0%

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

      7. associate-+l+ 82.0%

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

      8. metadata-eval 82.0%

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

      9. associate-+l+ 82.0%

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

      10. +-commutative 82.0%

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

      11. associate-+l+ 82.0%

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

      12. metadata-eval 82.0%

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

      13. +-commutative 82.0%

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

      14. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in alpha around 0 86.1%

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

   6. Taylor expanded in alpha around 0 80.8%

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

      1. +-commutative 80.8%

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

   8. Simplified 80.8%

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

   9. Taylor expanded in beta around inf 79.4%

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

       1. add0 79.4%

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

       2. *-commutative 79.4%

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

   11. Applied egg-rr 79.4%

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

       1. add0 79.4%

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

       2. associate-/r* 80.2%

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

       3. +-commutative 80.2%

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

   13. Simplified 80.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{2 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7) (/ 0.25 (+ alpha 3.0)) (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.25 / (alpha + 3.0);
	} 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 <= 2.7d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    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 <= 2.7) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = 0.25 / (alpha + 3.0)
	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 <= 2.7)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	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 <= 2.7)
		tmp = 0.25 / (alpha + 3.0);
	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, 2.7], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7000000000000002

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.0%

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

   6. Taylor expanded in alpha around 0 65.3%

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

   7. Taylor expanded in beta around 0 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]
   8. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{\alpha + 3}} \]

   if 2.7000000000000002 < beta

   1. Initial program 87.9%

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

   3. Simplified 90.6%

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

      1. associate-*l/ 90.5%

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

      2. +-commutative 90.5%

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

      3. div-inv 90.5%

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

      4. pow2 90.5%

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

      5. associate-+r+ 90.5%

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

      6. metadata-eval 90.5%

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

      7. pow-flip 91.5%

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

      8. metadata-eval 91.5%

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

      9. associate-+r+ 91.5%

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

      10. metadata-eval 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in beta around inf 82.9%

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

   8. Taylor expanded in alpha around 0 79.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4) (/ 0.25 (+ alpha 3.0)) (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} 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 <= 2.4d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    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 <= 2.4) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.25 / (alpha + 3.0)
	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 <= 2.4)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	else
		tmp = Float64(Float64(1.0 / 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 <= 2.4)
		tmp = 0.25 / (alpha + 3.0);
	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, 2.4], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-+l+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

      12. metadata-eval 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.0%

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

   6. Taylor expanded in alpha around 0 65.3%

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

   7. Taylor expanded in beta around 0 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]
   8. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{0.25}{\alpha + 3}} \]

   if 2.39999999999999991 < beta

   1. Initial program 87.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 90.5%

         \[\leadsto \color{blue}{\frac{\left(\beta + 1\right) \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}{\alpha + \left(\beta + 3\right)}} \]

      2. +-commutative 90.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}{\alpha + \left(\beta + 3\right)} \]

      3. div-inv 90.5%

         \[\leadsto \frac{\left(1 + \beta\right) \cdot \color{blue}{\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]

      4. pow2 90.5%

         \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot \frac{1}{\color{blue}{{\left(\alpha + \left(\beta + 2\right)\right)}^{2}}}\right)}{\alpha + \left(\beta + 3\right)} \]

      5. associate-+r+ 90.5%

         \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot \frac{1}{{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}}^{2}}\right)}{\alpha + \left(\beta + 3\right)} \]

      6. metadata-eval 90.5%

         \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot \frac{1}{{\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right)}^{2}}\right)}{\alpha + \left(\beta + 3\right)} \]

      7. pow-flip 91.5%

         \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot \color{blue}{{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}^{\left(-2\right)}}\right)}{\alpha + \left(\beta + 3\right)} \]

      8. metadata-eval 91.5%

         \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot {\left(\left(\alpha + \beta\right) + \color{blue}{2}\right)}^{\left(-2\right)}\right)}{\alpha + \left(\beta + 3\right)} \]

      9. associate-+r+ 91.5%

         \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot {\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}}^{\left(-2\right)}\right)}{\alpha + \left(\beta + 3\right)} \]

      10. metadata-eval 91.5%

          \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot {\left(\alpha + \left(\beta + 2\right)\right)}^{\color{blue}{-2}}\right)}{\alpha + \left(\beta + 3\right)} \]

   6. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\frac{\left(1 + \beta\right) \cdot \left(\left(\alpha + 1\right) \cdot {\left(\alpha + \left(\beta + 2\right)\right)}^{-2}\right)}{\alpha + \left(\beta + 3\right)}} \]

   7. Taylor expanded in beta around inf 82.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\alpha + \left(\beta + 3\right)} \]

   8. Taylor expanded in alpha around 0 79.3%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 80.1%

         \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{3 + \beta}} \]

      2. +-commutative 80.1%

         \[\leadsto \frac{\frac{1}{\beta}}{\color{blue}{\beta + 3}} \]

   10. Simplified 80.1%

       \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{\beta + 3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\alpha + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ alpha 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (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
    code = 0.25d0 / (alpha + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (alpha + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (alpha + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(alpha + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (alpha + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 94.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative 94.0%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. associate-+l+ 94.0%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. *-commutative 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. +-commutative 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. metadata-eval 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\beta + \alpha\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. associate-+l+ 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\beta + \left(\alpha + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. metadata-eval 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. associate-+l+ 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. +-commutative 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\color{blue}{\left(\beta + \alpha\right)} + \left(2 + 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. associate-+l+ 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\color{blue}{\left(\left(\left(\beta + \alpha\right) + 2\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. metadata-eval 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   13. +-commutative 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1\right)} \]

   14. metadata-eval 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\beta + \alpha\right) + \color{blue}{2}\right)} \]

3. Simplified 94.0%

   \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in alpha around 0 84.7%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)} \]

6. Taylor expanded in alpha around 0 71.5%

   \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]

7. Taylor expanded in beta around 0 44.9%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]
8. Step-by-step derivation

   1. +-commutative 44.9%

      \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]

9. Simplified 44.9%

   \[\leadsto \color{blue}{\frac{0.25}{\alpha + 3}} \]

10. Final simplification 44.9%

    \[\leadsto \frac{0.25}{\alpha + 3} \]
11. Add Preprocessing

Alternative 16: 45.8% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 95.9%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 94.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative 94.0%

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. associate-+l+ 94.0%

      \[\leadsto \frac{\frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. *-commutative 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. +-commutative 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. metadata-eval 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\beta + \alpha\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. associate-+l+ 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\beta + \left(\alpha + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. metadata-eval 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. associate-+l+ 94.0%

      \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. +-commutative 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\color{blue}{\left(\beta + \alpha\right)} + \left(2 + 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. associate-+l+ 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\color{blue}{\left(\left(\left(\beta + \alpha\right) + 2\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. metadata-eval 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   13. +-commutative 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1\right)} \]

   14. metadata-eval 94.0%

       \[\leadsto \frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\left(\beta + \alpha\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\beta + \alpha\right) + \color{blue}{2}\right)} \]

3. Simplified 94.0%

   \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\beta + \left(\alpha + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in alpha around 0 84.7%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)} \]

6. Taylor expanded in alpha around 0 68.7%

   \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
7. Step-by-step derivation

   1. +-commutative 68.7%

      \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \color{blue}{\left(\beta + 3\right)}} \]

8. Simplified 68.7%

   \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}} \]

9. Taylor expanded in beta around 0 42.9%

   \[\leadsto \color{blue}{0.08333333333333333} \]

10. Final simplification 42.9%

    \[\leadsto 0.08333333333333333 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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)))