Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.6%

Time: 13.7s

Alternatives: 20

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% 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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{t\_1}}{t\_1}}{1 + t\_1} \leq 0.1:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \beta\right) + 3}}{\frac{t\_0}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (+ (+ alpha beta) 2.0)))
   (if (<=
        (/
         (/ (/ (+ (+ (+ alpha beta) (* alpha beta)) 1.0) t_1) t_1)
         (+ 1.0 t_1))
        0.1)
     (/
      (/ 1.0 (+ (+ alpha beta) 3.0))
      (/ t_0 (/ (* (+ alpha 1.0) (+ beta 1.0)) t_0)))
     (* (/ alpha beta) (/ 1.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + beta) + 2.0;
	double tmp;
	if (((((((alpha + beta) + (alpha * beta)) + 1.0) / t_1) / t_1) / (1.0 + t_1)) <= 0.1) {
		tmp = (1.0 / ((alpha + beta) + 3.0)) / (t_0 / (((alpha + 1.0) * (beta + 1.0)) / t_0));
	} else {
		tmp = (alpha / beta) * (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 + (beta + 2.0d0)
    t_1 = (alpha + beta) + 2.0d0
    if (((((((alpha + beta) + (alpha * beta)) + 1.0d0) / t_1) / t_1) / (1.0d0 + t_1)) <= 0.1d0) then
        tmp = (1.0d0 / ((alpha + beta) + 3.0d0)) / (t_0 / (((alpha + 1.0d0) * (beta + 1.0d0)) / t_0))
    else
        tmp = (alpha / beta) * (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 + (beta + 2.0);
	double t_1 = (alpha + beta) + 2.0;
	double tmp;
	if (((((((alpha + beta) + (alpha * beta)) + 1.0) / t_1) / t_1) / (1.0 + t_1)) <= 0.1) {
		tmp = (1.0 / ((alpha + beta) + 3.0)) / (t_0 / (((alpha + 1.0) * (beta + 1.0)) / t_0));
	} else {
		tmp = (alpha / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (alpha + beta) + 2.0
	tmp = 0
	if ((((((alpha + beta) + (alpha * beta)) + 1.0) / t_1) / t_1) / (1.0 + t_1)) <= 0.1:
		tmp = (1.0 / ((alpha + beta) + 3.0)) / (t_0 / (((alpha + 1.0) * (beta + 1.0)) / t_0))
	else:
		tmp = (alpha / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(alpha * beta)) + 1.0) / t_1) / t_1) / Float64(1.0 + t_1)) <= 0.1)
		tmp = Float64(Float64(1.0 / Float64(Float64(alpha + beta) + 3.0)) / Float64(t_0 / Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / t_0)));
	else
		tmp = Float64(Float64(alpha / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (((((((alpha + beta) + (alpha * beta)) + 1.0) / t_1) / t_1) / (1.0 + t_1)) <= 0.1)
		tmp = (1.0 / ((alpha + beta) + 3.0)) / (t_0 / (((alpha + 1.0) * (beta + 1.0)) / t_0));
	else
		tmp = (alpha / beta) * (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[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(1.0 / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 0.10000000000000001

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 90.8%

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

   6. Applied egg-rr 99.8%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 1.6%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 49.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 49.4%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\left(\frac{1}{\beta}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + 1}{\beta}\right), \left(\frac{1}{\beta}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \left(\frac{\color{blue}{1}}{\beta}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \left(\frac{1}{\beta}\right)\right) \]

      7. /-lowering-/.f64 42.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \mathsf{/.f64}\left(1, \color{blue}{\beta}\right)\right) \]

   9. Applied egg-rr 42.5%

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

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\alpha}{\beta}\right)}, \mathsf{/.f64}\left(1, \beta\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 42.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\color{blue}{1}, \beta\right)\right) \]

   12. Simplified 42.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)} \leq 0.1:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \beta\right) + 3}}{\frac{\alpha + \left(\beta + 2\right)}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\alpha + \left(\beta + 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 3\right) \cdot t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+94)
     (/ (* (+ alpha 1.0) (/ (+ beta 1.0) (* (+ (+ alpha beta) 3.0) t_0))) t_0)
     (/ (/ (+ alpha 1.0) (+ beta (+ alpha 2.0))) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+94) {
		tmp = ((alpha + 1.0) * ((beta + 1.0) / (((alpha + beta) + 3.0) * t_0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5d+94) then
        tmp = ((alpha + 1.0d0) * ((beta + 1.0d0) / (((alpha + beta) + 3.0d0) * t_0))) / t_0
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+94) {
		tmp = ((alpha + 1.0) * ((beta + 1.0) / (((alpha + beta) + 3.0) * t_0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+94:
		tmp = ((alpha + 1.0) * ((beta + 1.0) / (((alpha + beta) + 3.0) * t_0))) / t_0
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+94)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(beta + 1.0) / Float64(Float64(Float64(alpha + beta) + 3.0) * t_0))) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+94)
		tmp = ((alpha + 1.0) * ((beta + 1.0) / (((alpha + beta) + 3.0) * t_0))) / t_0;
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+94], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0000000000000001e94

   1. Initial program 98.9%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.3%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right) \]

   6. Applied egg-rr 99.2%

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

   if 5.0000000000000001e94 < beta

   1. Initial program 73.7%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 84.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 84.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right), \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right), \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\beta + 2\right) + \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + \left(2 + \alpha\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \left(2 + \alpha\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right) \]

      13. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + \color{blue}{\left(2 + 1\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + 3\right)\right) \]

      15. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\alpha + \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      17. +-lowering-+.f64 84.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Applied egg-rr 84.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{t\_0 \cdot \left(\left(\left(\alpha + \beta\right) + 3\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 5e+94)
     (/ (* (+ alpha 1.0) (+ beta 1.0)) (* t_0 (* (+ (+ alpha beta) 3.0) t_0)))
     (/ (/ (+ alpha 1.0) (+ beta (+ alpha 2.0))) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 5e+94) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * (((alpha + beta) + 3.0) * t_0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 5d+94) then
        tmp = ((alpha + 1.0d0) * (beta + 1.0d0)) / (t_0 * (((alpha + beta) + 3.0d0) * t_0))
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 5e+94) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * (((alpha + beta) + 3.0) * t_0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 5e+94:
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * (((alpha + beta) + 3.0) * t_0))
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 5e+94)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / Float64(t_0 * Float64(Float64(Float64(alpha + beta) + 3.0) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 5e+94)
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * (((alpha + beta) + 3.0) * t_0));
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+94], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0000000000000001e94

   1. Initial program 98.9%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.3%

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

   if 5.0000000000000001e94 < beta

   1. Initial program 73.7%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 84.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 84.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right), \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right), \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\beta + 2\right) + \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + \left(2 + \alpha\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \left(2 + \alpha\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right) \]

      13. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + \color{blue}{\left(2 + 1\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + 3\right)\right) \]

      15. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\alpha + \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      17. +-lowering-+.f64 84.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Applied egg-rr 84.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 170000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 170000000.0)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/
    (* (+ alpha 1.0) (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))
    (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 170000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 170000000.0d0) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 170000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 170000000.0:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 170000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta)) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 170000000.0)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha + 1.0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 170000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.7e8

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.0%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.2%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{\beta + 2}\right), \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(\beta + 2\right)\right), \left(\color{blue}{\left(\beta + 2\right)} \cdot \left(\beta + 3\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\beta + 2\right)\right), \left(\left(\color{blue}{\beta} + 2\right) \cdot \left(\beta + 3\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(\beta + \color{blue}{2}\right) \cdot \left(\beta + 3\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\color{blue}{\beta} + 3\right)\right)\right) \]

      9. +-lowering-+.f64 67.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   9. Applied egg-rr 67.2%

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

   if 1.7e8 < beta

   1. Initial program 80.6%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 65.6%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right) \]

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \color{blue}{\left(\frac{1 + -1 \cdot \frac{4 + 2 \cdot \alpha}{\beta}}{\beta}\right)}\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{4 + 2 \cdot \alpha}{\beta}\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-1 \cdot \frac{4 + 2 \cdot \alpha}{\beta}\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-1, \left(\frac{4 + 2 \cdot \alpha}{\beta}\right)\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(4 + 2 \cdot \alpha\right), \beta\right)\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(2 \cdot \alpha\right)\right), \beta\right)\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      6. *-lowering-*.f64 79.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(2, \alpha\right)\right), \beta\right)\right)\right), \beta\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 79.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 170000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+18}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta + \left(\alpha + 2\right)}{\alpha + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+18)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/
    (/ 1.0 (+ alpha (+ beta 3.0)))
    (/ (+ beta (+ alpha 2.0)) (+ alpha 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+18) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / ((beta + (alpha + 2.0)) / (alpha + 1.0));
	}
	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 <= 1d+18) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = (1.0d0 / (alpha + (beta + 3.0d0))) / ((beta + (alpha + 2.0d0)) / (alpha + 1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+18) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / ((beta + (alpha + 2.0)) / (alpha + 1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1e+18:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = (1.0 / (alpha + (beta + 3.0))) / ((beta + (alpha + 2.0)) / (alpha + 1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+18)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 3.0))) / Float64(Float64(beta + Float64(alpha + 2.0)) / Float64(alpha + 1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+18)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = (1.0 / (alpha + (beta + 3.0))) / ((beta + (alpha + 2.0)) / (alpha + 1.0));
	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, 1e+18], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e18

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.1%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.6%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{\beta + 2}\right), \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(\beta + 2\right)\right), \left(\color{blue}{\left(\beta + 2\right)} \cdot \left(\beta + 3\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\beta + 2\right)\right), \left(\left(\color{blue}{\beta} + 2\right) \cdot \left(\beta + 3\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(\beta + \color{blue}{2}\right) \cdot \left(\beta + 3\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\color{blue}{\beta} + 3\right)\right)\right) \]

      9. +-lowering-+.f64 67.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   9. Applied egg-rr 67.6%

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

   if 1e18 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 80.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 80.5%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \alpha}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + 3\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\alpha + \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\alpha + \color{blue}{1}}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right), \color{blue}{\left(\alpha + 1\right)}\right)\right) \]

   7. Applied egg-rr 80.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+18}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta + \left(\alpha + 2\right)}{\alpha + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.6e+14)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) (+ beta (+ alpha 2.0))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.6e+14) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.6d+14) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.6e+14) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.6e+14:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.6e+14)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.6e+14)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.6e+14], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.6e14

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.1%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.6%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{\beta + 2}\right), \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(\beta + 2\right)\right), \left(\color{blue}{\left(\beta + 2\right)} \cdot \left(\beta + 3\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\beta + 2\right)\right), \left(\left(\color{blue}{\beta} + 2\right) \cdot \left(\beta + 3\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(\beta + \color{blue}{2}\right) \cdot \left(\beta + 3\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\color{blue}{\beta} + 3\right)\right)\right) \]

      9. +-lowering-+.f64 67.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   9. Applied egg-rr 67.6%

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

   if 9.6e14 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 80.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 80.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right), \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right), \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\alpha + \beta\right) + 2\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\left(\beta + 2\right) + \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + \left(2 + \alpha\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \left(2 + \alpha\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}\right) + 1\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right) \]

      13. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + \color{blue}{\left(2 + 1\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + 3\right)\right) \]

      15. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\alpha + \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\alpha, \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      17. +-lowering-+.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Applied egg-rr 80.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9e+15)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ 1.0 (+ alpha (+ beta 3.0))) (/ beta (+ alpha 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9e+15) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	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 <= 3.9d+15) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = (1.0d0 / (alpha + (beta + 3.0d0))) / (beta / (alpha + 1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9e+15) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.9e+15:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.9e+15)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 3.0))) / Float64(beta / Float64(alpha + 1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.9e+15)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	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, 3.9e+15], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.9e15

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.1%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.6%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{\beta + 2}\right), \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(\beta + 2\right)\right), \left(\color{blue}{\left(\beta + 2\right)} \cdot \left(\beta + 3\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\beta + 2\right)\right), \left(\left(\color{blue}{\beta} + 2\right) \cdot \left(\beta + 3\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(\beta + \color{blue}{2}\right) \cdot \left(\beta + 3\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\color{blue}{\beta} + 3\right)\right)\right) \]

      9. +-lowering-+.f64 67.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   9. Applied egg-rr 67.6%

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

   if 3.9e15 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 80.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 80.5%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \alpha}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + 3\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\alpha + \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\alpha + \color{blue}{1}}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right), \color{blue}{\left(\alpha + 1\right)}\right)\right) \]

   7. Applied egg-rr 80.5%

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

   8. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \color{blue}{\left(\frac{\beta}{1 + \alpha}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\beta, \color{blue}{\left(1 + \alpha\right)}\right)\right) \]

      2. +-lowering-+.f64 80.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(1, \color{blue}{\alpha}\right)\right)\right) \]

   10. Simplified 80.0%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2e+17)
   (/ (+ beta 1.0) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ 1.0 (+ alpha (+ beta 3.0))) (/ beta (+ alpha 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e+17) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	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 <= 1.2d+17) then
        tmp = (beta + 1.0d0) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = (1.0d0 / (alpha + (beta + 3.0d0))) / (beta / (alpha + 1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e+17) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2e+17:
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2e+17)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 3.0))) / Float64(beta / Float64(alpha + 1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2e+17)
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	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, 1.2e+17], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.2e17

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 94.1%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.6%

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

   if 1.2e17 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 80.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 80.5%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \alpha}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + 3\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\alpha + \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\alpha + \color{blue}{1}}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right), \color{blue}{\left(\alpha + 1\right)}\right)\right) \]

   7. Applied egg-rr 80.5%

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

   8. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \color{blue}{\left(\frac{\beta}{1 + \alpha}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\beta, \color{blue}{\left(1 + \alpha\right)}\right)\right) \]

      2. +-lowering-+.f64 80.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(1, \color{blue}{\alpha}\right)\right)\right) \]

   10. Simplified 80.0%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (*
     beta
     (+
      (* beta (+ (* beta 0.024691358024691357) -0.011574074074074073))
      -0.027777777777777776)))
   (/ (/ 1.0 (+ alpha (+ beta 3.0))) (/ beta (+ alpha 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	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 <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) + (-0.011574074074074073d0))) + (-0.027777777777777776d0)))
    else
        tmp = (1.0d0 / (alpha + (beta + 3.0d0))) / (beta / (alpha + 1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776))
	else:
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776)));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 3.0))) / Float64(beta / Float64(alpha + 1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	else
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	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, 1.7], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(beta * 0.024691358024691357), $MachinePrecision] + -0.011574074074074073), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)\right)\right), \frac{-1}{36}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{2}{81} \cdot \beta\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      11. *-lowering-*.f64 66.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 66.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)} \]

   if 1.69999999999999996 < beta

   1. Initial program 81.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 77.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 77.4%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \alpha}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2\right) + 1\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{1 + \alpha}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\alpha + \beta\right) + 3\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\alpha + \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \left(\beta + 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}{1 + \alpha}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot \color{blue}{1}}{1 + \alpha}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\alpha + \color{blue}{1}}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right), \color{blue}{\left(\alpha + 1\right)}\right)\right) \]

   7. Applied egg-rr 77.4%

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

   8. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \color{blue}{\left(\frac{\beta}{1 + \alpha}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\beta, \color{blue}{\left(1 + \alpha\right)}\right)\right) \]

      2. +-lowering-+.f64 76.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(1, \color{blue}{\alpha}\right)\right)\right) \]

   10. Simplified 76.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8)
   (+
    0.08333333333333333
    (*
     beta
     (+
      (* beta (+ (* beta 0.024691358024691357) -0.011574074074074073))
      -0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.8d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) + (-0.011574074074074073d0))) + (-0.027777777777777776d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.8:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.8)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.8], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(beta * 0.024691358024691357), $MachinePrecision] + -0.011574074074074073), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.80000000000000004

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)\right)\right), \frac{-1}{36}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{2}{81} \cdot \beta\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      11. *-lowering-*.f64 66.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 66.6%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)} \]

   if 1.80000000000000004 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right) \]

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 76.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 76.6%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.85)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 5e+158)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 5e+158) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.85d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 5d+158) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.85) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 5e+158) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.85:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 5e+158:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.85)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 5e+158)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.85)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 5e+158)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.85], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5e+158], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.85000000000000009

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.85000000000000009 < beta < 4.9999999999999996e158

   1. Initial program 88.6%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 65.7%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 66.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 66.4%

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

   if 4.9999999999999996e158 < beta

   1. Initial program 71.9%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 68.5%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 89.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 89.1%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\beta}\right), \beta\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 89.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 89.2%

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

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\alpha}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 88.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\alpha, \beta\right), \beta\right) \]

   12. Simplified 88.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.6:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.6)
   (+
    0.08333333333333333
    (* beta (+ -0.027777777777777776 (* beta -0.011574074074074073))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.6) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.6d0) then
        tmp = 0.08333333333333333d0 + (beta * ((-0.027777777777777776d0) + (beta * (-0.011574074074074073d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.6) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.6:
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.6)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(-0.027777777777777776 + Float64(beta * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.6)
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.6], N[(0.08333333333333333 + N[(beta * N[(-0.027777777777777776 + N[(beta * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6000000000000001

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{-5}{432} \cdot \beta\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

      7. *-lowering-*.f64 66.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 66.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 1.6000000000000001 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}\right) \]

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 76.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 76.6%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.6:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 1.46 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 1.46e+161) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 1.46e+161) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 1.46d+161) then
        tmp = (1.0d0 / beta) / beta
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 1.46e+161) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 1.46e+161:
		tmp = (1.0 / beta) / beta
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 1.46e+161)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 1.46e+161)
		tmp = (1.0 / beta) / beta;
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.46e+161], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999998

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta < 1.46000000000000008e161

   1. Initial program 87.0%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 64.8%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 65.5%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\beta}\right), \beta\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 65.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 65.7%

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

   10. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 51.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \beta\right), \beta\right) \]

   12. Simplified 51.2%

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

   if 1.46000000000000008e161 < beta

   1. Initial program 73.2%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 70.0%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 91.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 91.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\beta}\right), \beta\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 91.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 91.4%

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

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\alpha}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 91.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\alpha, \beta\right), \beta\right) \]

   12. Simplified 91.4%

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

Alternative 14: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.65)
   (+
    0.08333333333333333
    (* beta (+ -0.027777777777777776 (* beta -0.011574074074074073))))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.65d0) then
        tmp = 0.08333333333333333d0 + (beta * ((-0.027777777777777776d0) + (beta * (-0.011574074074074073d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.65:
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.65)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(-0.027777777777777776 + Float64(beta * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.65)
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.65], N[(0.08333333333333333 + N[(beta * N[(-0.027777777777777776 + N[(beta * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6499999999999999

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{-5}{432} \cdot \beta\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

      7. *-lowering-*.f64 66.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 66.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 1.6499999999999999 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 76.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\beta}\right), \beta\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 76.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 76.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\beta}\right), \beta\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 76.3%

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

Alternative 16: 91.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ 1.0 beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = (1.0 / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (1.0 / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 76.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + 1}{\beta}\right), \beta\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 76.3%

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

   10. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{\beta}\right)}, \beta\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 67.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \beta\right), \beta\right) \]

   12. Simplified 67.9%

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

Alternative 17: 91.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 76.3%

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

   8. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      3. *-lowering-*.f64 67.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   10. Simplified 67.7%

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

Alternative 18: 47.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.14285714285714285}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.14285714285714285 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.14285714285714285 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.14285714285714285d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.14285714285714285 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.14285714285714285 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.14285714285714285 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.14285714285714285 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.14285714285714285 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   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/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.2999999999999998 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 64.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 64.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \color{blue}{\left(12 + \beta \cdot \left(16 + 7 \cdot \beta\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \color{blue}{\left(\beta \cdot \left(16 + 7 \cdot \beta\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \color{blue}{\left(16 + 7 \cdot \beta\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \color{blue}{\left(7 \cdot \beta\right)}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \left(\beta \cdot \color{blue}{7}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 42.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \color{blue}{7}\right)\right)\right)\right)\right) \]

   10. Simplified 42.9%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot 7\right)}} \]

   11. Taylor expanded in beta around inf

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

       1. /-lowering-/.f64 7.4%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{7}, \color{blue}{\beta}\right) \]

   13. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{0.14285714285714285}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.14285714285714285}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7) 0.08333333333333333 (/ 0.14285714285714285 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.14285714285714285 / 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 <= 1.7d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.14285714285714285d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.14285714285714285 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333
	else:
		tmp = 0.14285714285714285 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.14285714285714285 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333;
	else
		tmp = 0.14285714285714285 / 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, 1.7], 0.08333333333333333, N[(0.14285714285714285 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 93.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 67.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12}} \]
   9. Step-by-step derivation

   10. Simplified 65.3%

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

   if 1.69999999999999996 < beta

   1. Initial program 81.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 64.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 64.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \color{blue}{\left(12 + \beta \cdot \left(16 + 7 \cdot \beta\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \color{blue}{\left(\beta \cdot \left(16 + 7 \cdot \beta\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \color{blue}{\left(16 + 7 \cdot \beta\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \color{blue}{\left(7 \cdot \beta\right)}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \left(\beta \cdot \color{blue}{7}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 42.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(16, \mathsf{*.f64}\left(\beta, \color{blue}{7}\right)\right)\right)\right)\right) \]

   10. Simplified 42.9%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot 7\right)}} \]

   11. Taylor expanded in beta around inf

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

       1. /-lowering-/.f64 7.4%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{7}, \color{blue}{\beta}\right) \]

   13. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{0.14285714285714285}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.9% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l/ N/A

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

   2. associate-/l/ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\beta \cdot \alpha + \left(\alpha + \left(\beta + 1\right)\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   8. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \alpha + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\alpha \cdot \left(\beta + 1\right) + \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(\alpha + 1\right) \cdot \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(\beta + 1\right)\right), \left(\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\alpha + 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\beta + 1\right)\right), \left(\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\beta, 1\right)\right), \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \color{blue}{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)\right)\right) \]

3. Simplified 85.5%

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

5. Taylor expanded in alpha around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

   11. +-lowering-+.f64 66.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

7. Simplified 66.8%

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

8. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1}{12}} \]
9. Step-by-step derivation

10. Simplified 46.2%

    \[\leadsto \color{blue}{0.08333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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)))