Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.4%

Time: 13.0s

Alternatives: 16

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 2.12 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t\_0 \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 2.12e+36)
     (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
     (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2.12e+36) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 2.12d+36) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / (t_0 * (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2.12e+36) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 2.12e+36:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 2.12e+36)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 2.12e+36)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.12e+36], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.12000000000000004e36

   1. Initial program 99.9%

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

   3. Simplified 93.5%

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

   if 2.12000000000000004e36 < beta

   1. Initial program 80.9%

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

   3. Taylor expanded in beta around inf 89.5%

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

   4. Taylor expanded in alpha around 0 89.5%

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

      1. associate-+r+ 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.12 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Simplified 85.3%

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

   1. associate-+r+ 85.3%

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

   2. fma-undefine 85.3%

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

   3. *-commutative 85.3%

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

   4. associate-+l+ 85.3%

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

   5. +-commutative 85.3%

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

   6. associate-+l+ 85.3%

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

   7. *-commutative 85.3%

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

   8. associate-*r* 85.3%

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

   9. associate-+r+ 85.3%

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

   10. +-commutative 85.3%

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

   11. associate-/l/ 93.9%

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

   12. clear-num 93.9%

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

   13. inv-pow 93.9%

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

6. Applied egg-rr 93.9%

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

   1. unpow-1 93.9%

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

   2. associate-/l* 94.4%

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

   3. associate-+r+ 94.4%

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

   4. +-commutative 94.4%

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

   5. +-commutative 94.4%

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

   6. associate-+r+ 94.4%

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

   7. +-commutative 94.4%

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

   8. associate-+r+ 94.4%

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

   9. +-commutative 94.4%

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

   10. fma-undefine 94.4%

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

   11. *-commutative 94.4%

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

   12. +-commutative 94.4%

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

   13. associate-+r+ 94.4%

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

   14. +-commutative 94.4%

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

   15. *-rgt-identity 94.4%

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

   16. *-commutative 94.4%

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

   17. +-commutative 94.4%

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

   18. distribute-lft-in 94.4%

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

   19. +-commutative 94.4%

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

   20. *-commutative 94.4%

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

8. Simplified 94.4%

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

   1. associate-/l* 98.7%

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3e+27)
   (/
    1.0
    (* (+ 2.0 (+ beta alpha)) (* (+ 2.0 beta) (/ (+ beta 3.0) (+ 1.0 beta)))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+27) {
		tmp = 1.0 / ((2.0 + (beta + alpha)) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3d+27) then
        tmp = 1.0d0 / ((2.0d0 + (beta + alpha)) * ((2.0d0 + beta) * ((beta + 3.0d0) / (1.0d0 + beta))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+27) {
		tmp = 1.0 / ((2.0 + (beta + alpha)) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3e+27:
		tmp = 1.0 / ((2.0 + (beta + alpha)) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3e+27)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(beta + alpha)) * Float64(Float64(2.0 + beta) * Float64(Float64(beta + 3.0) / Float64(1.0 + beta)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3e+27)
		tmp = 1.0 / ((2.0 + (beta + alpha)) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3e+27], N[(1.0 / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.99999999999999976e27

   1. Initial program 99.9%

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

   3. Simplified 93.4%

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

      1. associate-+r+ 93.4%

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

      2. fma-undefine 93.4%

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

      3. *-commutative 93.4%

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

      4. associate-+l+ 93.4%

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

      5. +-commutative 93.4%

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

      6. associate-+l+ 93.4%

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

      7. *-commutative 93.4%

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

      8. associate-*r* 93.4%

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

      9. associate-+r+ 93.4%

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

      10. +-commutative 93.4%

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

      11. associate-/l/ 99.2%

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

      12. clear-num 99.2%

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

      13. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. associate-+r+ 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-+r+ 99.2%

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

      7. +-commutative 99.2%

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

      8. associate-+r+ 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. *-commutative 99.2%

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

      12. +-commutative 99.2%

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

      13. associate-+r+ 99.2%

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

      14. +-commutative 99.2%

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

      15. *-rgt-identity 99.2%

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

      16. *-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. distribute-lft-in 99.2%

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

      19. +-commutative 99.2%

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

      20. *-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in alpha around 0 71.0%

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

       1. associate-/l* 71.0%

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

   11. Simplified 71.0%

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

   if 2.99999999999999976e27 < beta

   1. Initial program 81.4%

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

   3. Taylor expanded in beta around inf 89.8%

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

   4. Taylor expanded in alpha around 0 89.8%

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

      1. associate-+r+ 89.8%

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

   6. Simplified 89.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.45 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.45e+27)
   (/ 1.0 (* (+ 2.0 beta) (* (+ 2.0 beta) (/ (+ beta 3.0) (+ 1.0 beta)))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.45e+27) {
		tmp = 1.0 / ((2.0 + beta) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.45d+27) then
        tmp = 1.0d0 / ((2.0d0 + beta) * ((2.0d0 + beta) * ((beta + 3.0d0) / (1.0d0 + beta))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.45e+27) {
		tmp = 1.0 / ((2.0 + beta) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.45e+27:
		tmp = 1.0 / ((2.0 + beta) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.45e+27)
		tmp = Float64(1.0 / Float64(Float64(2.0 + beta) * Float64(Float64(2.0 + beta) * Float64(Float64(beta + 3.0) / Float64(1.0 + beta)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.45e+27)
		tmp = 1.0 / ((2.0 + beta) * ((2.0 + beta) * ((beta + 3.0) / (1.0 + beta))));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.45e+27], N[(1.0 / N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.45000000000000009e27

   1. Initial program 99.9%

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

   3. Simplified 93.4%

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

      1. associate-+r+ 93.4%

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

      2. fma-undefine 93.4%

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

      3. *-commutative 93.4%

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

      4. associate-+l+ 93.4%

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

      5. +-commutative 93.4%

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

      6. associate-+l+ 93.4%

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

      7. *-commutative 93.4%

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

      8. associate-*r* 93.4%

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

      9. associate-+r+ 93.4%

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

      10. +-commutative 93.4%

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

      11. associate-/l/ 99.2%

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

      12. clear-num 99.2%

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

      13. inv-pow 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. associate-+r+ 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-+r+ 99.2%

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

      7. +-commutative 99.2%

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

      8. associate-+r+ 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-undefine 99.2%

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

      11. *-commutative 99.2%

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

      12. +-commutative 99.2%

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

      13. associate-+r+ 99.2%

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

      14. +-commutative 99.2%

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

      15. *-rgt-identity 99.2%

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

      16. *-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. distribute-lft-in 99.2%

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

      19. +-commutative 99.2%

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

      20. *-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in alpha around 0 71.0%

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

       1. associate-/l* 71.0%

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

   11. Simplified 71.0%

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

   12. Taylor expanded in alpha around 0 70.1%

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

   if 3.45000000000000009e27 < beta

   1. Initial program 81.4%

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

   3. Taylor expanded in beta around inf 89.8%

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

   4. Taylor expanded in alpha around 0 89.8%

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

      1. associate-+r+ 89.8%

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

   6. Simplified 89.8%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.45 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \alpha}}{\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta}}{\beta \cdot \left(\frac{-3 - \alpha}{\beta} + -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6)
   (/ (/ (+ 1.0 alpha) (+ 2.0 alpha)) (* (+ 2.0 alpha) (+ alpha 3.0)))
   (/ (/ (- -1.0 alpha) beta) (* beta (+ (/ (- -3.0 alpha) beta) -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0));
	} else {
		tmp = ((-1.0 - alpha) / beta) / (beta * (((-3.0 - alpha) / beta) + -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 <= 3.6d0) then
        tmp = ((1.0d0 + alpha) / (2.0d0 + alpha)) / ((2.0d0 + alpha) * (alpha + 3.0d0))
    else
        tmp = (((-1.0d0) - alpha) / beta) / (beta * ((((-3.0d0) - alpha) / beta) + (-1.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0));
	} else {
		tmp = ((-1.0 - alpha) / beta) / (beta * (((-3.0 - alpha) / beta) + -1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0))
	else:
		tmp = ((-1.0 - alpha) / beta) / (beta * (((-3.0 - alpha) / beta) + -1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + alpha)) / Float64(Float64(2.0 + alpha) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) / beta) / Float64(beta * Float64(Float64(Float64(-3.0 - alpha) / beta) + -1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((2.0 + alpha) * (alpha + 3.0));
	else
		tmp = ((-1.0 - alpha) / beta) / (beta * (((-3.0 - alpha) / beta) + -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, 3.6], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta * N[(N[(N[(-3.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   1. Initial program 99.9%

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

      1. associate-/l/ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.1%

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

      4. *-commutative 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-+l+ 99.1%

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

      7. metadata-eval 99.1%

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

      8. +-commutative 99.1%

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

      9. +-commutative 99.1%

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

      10. +-commutative 99.1%

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

      11. metadata-eval 99.1%

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

      12. metadata-eval 99.1%

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

      13. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in beta around 0 97.2%

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

   6. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

      2. +-commutative 97.3%

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

   8. Simplified 97.3%

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

   if 3.60000000000000009 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in beta around -inf 85.6%

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

      1. mul-1-neg 85.6%

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

      2. distribute-rgt-neg-in 85.6%

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

      3. sub-neg 85.6%

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

      4. associate-*r/ 85.6%

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

      5. distribute-lft-in 85.6%

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

      6. metadata-eval 85.6%

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

      7. metadata-eval 85.6%

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

      8. mul-1-neg 85.6%

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

      9. unsub-neg 85.6%

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

      10. metadata-eval 85.6%

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

      11. metadata-eval 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \alpha}}{\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta}}{\beta \cdot \left(\frac{-3 - \alpha}{\beta} + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.9%

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

      1. associate-/l/ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.1%

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

      4. *-commutative 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-+l+ 99.1%

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

      7. metadata-eval 99.1%

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

      8. +-commutative 99.1%

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

      9. +-commutative 99.1%

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

      10. +-commutative 99.1%

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

      11. metadata-eval 99.1%

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

      12. metadata-eval 99.1%

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

      13. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in beta around 0 97.2%

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

   6. Taylor expanded in beta around 0 97.3%

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

      1. +-commutative 97.3%

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

      2. +-commutative 97.3%

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

   8. Simplified 97.3%

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

   if 3 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.6%

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

      1. associate-+r+ 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 93.9%

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

Alternative 7: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\frac{1 + \alpha}{12 + \alpha \cdot \left(16 + \alpha \cdot 7\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0)
   (/ (+ 1.0 alpha) (+ 12.0 (* alpha (+ 16.0 (* alpha 7.0)))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = (1.0 + alpha) / (12.0 + (alpha * (16.0 + (alpha * 7.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.0d0) then
        tmp = (1.0d0 + alpha) / (12.0d0 + (alpha * (16.0d0 + (alpha * 7.0d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = (1.0 + alpha) / (12.0 + (alpha * (16.0 + (alpha * 7.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.0:
		tmp = (1.0 + alpha) / (12.0 + (alpha * (16.0 + (alpha * 7.0))))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = Float64(Float64(1.0 + alpha) / Float64(12.0 + Float64(alpha * Float64(16.0 + Float64(alpha * 7.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.0)
		tmp = (1.0 + alpha) / (12.0 + (alpha * (16.0 + (alpha * 7.0))));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.0], N[(N[(1.0 + alpha), $MachinePrecision] / N[(12.0 + N[(alpha * N[(16.0 + N[(alpha * 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 83.4%

       \[\leadsto \frac{1 + \alpha}{\color{blue}{12 + \alpha \cdot \left(16 + 7 \cdot \alpha\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 83.4%

          \[\leadsto \frac{1 + \alpha}{12 + \alpha \cdot \left(16 + \color{blue}{\alpha \cdot 7}\right)} \]

   13. Simplified 83.4%

       \[\leadsto \frac{1 + \alpha}{\color{blue}{12 + \alpha \cdot \left(16 + \alpha \cdot 7\right)}} \]

   if 3 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.6%

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

      1. associate-+r+ 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\frac{1 + \alpha}{12 + \alpha \cdot \left(16 + \alpha \cdot 7\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (+
    0.08333333333333333
    (*
     alpha
     (-
      (* alpha (- (* alpha 0.024691358024691357) 0.011574074074074073))
      0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * ((alpha * 0.024691358024691357d0) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * Float64(Float64(alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * N[(N[(alpha * 0.024691358024691357), $MachinePrecision] - 0.011574074074074073), $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 69.0%

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

   if 2.2000000000000002 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.6%

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

      1. associate-+r+ 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+
    0.08333333333333333
    (* alpha (- (* alpha -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 68.8%

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

   if 2.10000000000000009 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.6%

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

      1. associate-+r+ 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (+
    0.08333333333333333
    (* alpha (- (* alpha -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 68.8%

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

   if 2.2000000000000002 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (+
    0.08333333333333333
    (* alpha (- (* alpha -0.011574074074074073) 0.027777777777777776)))
   (/ 1.0 (* beta (/ beta (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.4d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta / (1.0d0 + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.4:
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta / Float64(1.0 + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.4)
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 68.8%

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

   if 3.39999999999999991 < beta

   1. Initial program 83.9%

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

   3. Simplified 66.4%

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

      1. associate-+r+ 66.4%

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

      2. fma-undefine 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-+l+ 66.4%

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

      5. +-commutative 66.4%

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

      6. associate-+l+ 66.4%

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

      7. *-commutative 66.4%

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

      8. associate-*r* 66.3%

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

      9. associate-+r+ 66.3%

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

      10. +-commutative 66.3%

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

      11. associate-/l/ 81.2%

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

      12. clear-num 81.1%

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

      13. inv-pow 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. unpow-1 81.1%

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

      2. associate-/l* 82.7%

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

      3. associate-+r+ 82.7%

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

      4. +-commutative 82.7%

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

      5. +-commutative 82.7%

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

      6. associate-+r+ 82.7%

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

      7. +-commutative 82.7%

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

      8. associate-+r+ 82.7%

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

      9. +-commutative 82.7%

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

      10. fma-undefine 82.7%

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

      11. *-commutative 82.7%

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

      12. +-commutative 82.7%

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

      13. associate-+r+ 82.7%

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

      14. +-commutative 82.7%

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

      15. *-rgt-identity 82.7%

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

      16. *-commutative 82.7%

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

      17. +-commutative 82.7%

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

      18. distribute-lft-in 82.7%

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

      19. +-commutative 82.7%

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

      20. *-commutative 82.7%

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

   8. Simplified 82.7%

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

   9. Taylor expanded in beta around inf 83.5%

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

   10. Taylor expanded in beta around inf 83.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (+
    0.08333333333333333
    (* alpha (- (* alpha -0.011574074074074073) 0.027777777777777776)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 68.8%

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

   if 2.2000000000000002 < beta

   1. Initial program 83.9%

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

   3. Simplified 66.4%

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

      1. associate-+r+ 66.4%

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

      2. fma-undefine 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-+l+ 66.4%

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

      5. +-commutative 66.4%

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

      6. associate-+l+ 66.4%

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

      7. *-commutative 66.4%

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

      8. associate-*r* 66.3%

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

      9. associate-+r+ 66.3%

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

      10. +-commutative 66.3%

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

      11. associate-/l/ 81.2%

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

      12. clear-num 81.1%

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

      13. inv-pow 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. unpow-1 81.1%

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

      2. associate-/l* 82.7%

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

      3. associate-+r+ 82.7%

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

      4. +-commutative 82.7%

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

      5. +-commutative 82.7%

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

      6. associate-+r+ 82.7%

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

      7. +-commutative 82.7%

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

      8. associate-+r+ 82.7%

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

      9. +-commutative 82.7%

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

      10. fma-undefine 82.7%

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

      11. *-commutative 82.7%

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

      12. +-commutative 82.7%

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

      13. associate-+r+ 82.7%

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

      14. +-commutative 82.7%

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

      15. *-rgt-identity 82.7%

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

      16. *-commutative 82.7%

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

      17. +-commutative 82.7%

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

      18. distribute-lft-in 82.7%

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

      19. +-commutative 82.7%

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

      20. *-commutative 82.7%

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

   8. Simplified 82.7%

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

   9. Taylor expanded in beta around inf 84.0%

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

   10. Taylor expanded in alpha around 0 78.1%

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

       1. associate-/r* 78.5%

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

   12. Simplified 78.5%

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

   13. Taylor expanded in beta around inf 78.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 68.7%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
   12. Step-by-step derivation

       1. *-commutative 68.7%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

   13. Simplified 68.7%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 2.10000000000000009 < beta

   1. Initial program 83.9%

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

   3. Simplified 66.4%

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

      1. associate-+r+ 66.4%

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

      2. fma-undefine 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-+l+ 66.4%

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

      5. +-commutative 66.4%

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

      6. associate-+l+ 66.4%

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

      7. *-commutative 66.4%

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

      8. associate-*r* 66.3%

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

      9. associate-+r+ 66.3%

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

      10. +-commutative 66.3%

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

      11. associate-/l/ 81.2%

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

      12. clear-num 81.1%

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

      13. inv-pow 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. unpow-1 81.1%

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

      2. associate-/l* 82.7%

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

      3. associate-+r+ 82.7%

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

      4. +-commutative 82.7%

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

      5. +-commutative 82.7%

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

      6. associate-+r+ 82.7%

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

      7. +-commutative 82.7%

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

      8. associate-+r+ 82.7%

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

      9. +-commutative 82.7%

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

      10. fma-undefine 82.7%

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

      11. *-commutative 82.7%

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

      12. +-commutative 82.7%

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

      13. associate-+r+ 82.7%

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

      14. +-commutative 82.7%

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

      15. *-rgt-identity 82.7%

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

      16. *-commutative 82.7%

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

      17. +-commutative 82.7%

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

      18. distribute-lft-in 82.7%

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

      19. +-commutative 82.7%

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

      20. *-commutative 82.7%

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

   8. Simplified 82.7%

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

   9. Taylor expanded in beta around inf 84.0%

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

   10. Taylor expanded in alpha around 0 78.1%

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

       1. associate-/r* 78.5%

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

   12. Simplified 78.5%

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

   13. Taylor expanded in beta around inf 78.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. sub-neg 92.9%

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

      4. associate-*r/ 92.9%

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

      5. distribute-lft-in 92.9%

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

      6. metadata-eval 92.9%

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

      7. mul-1-neg 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in beta around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in alpha around 0 68.7%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
   12. Step-by-step derivation

       1. *-commutative 68.7%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

   13. Simplified 68.7%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]

   if 2 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 78.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 + \alpha \cdot -0.027777777777777776 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (+ 0.08333333333333333 (* alpha -0.027777777777777776)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333 + (alpha * -0.027777777777777776);
}

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 + (alpha * (-0.027777777777777776d0))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333 + (alpha * -0.027777777777777776);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333 + (alpha * -0.027777777777777776)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
end

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Simplified 85.3%

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

5. Taylor expanded in alpha around -inf 83.7%

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

   1. mul-1-neg 83.7%

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

   2. distribute-rgt-neg-in 83.7%

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

   3. sub-neg 83.7%

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

   4. associate-*r/ 83.7%

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

   5. distribute-lft-in 83.7%

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

   6. metadata-eval 83.7%

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

   7. mul-1-neg 83.7%

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

   8. unsub-neg 83.7%

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

   9. metadata-eval 83.7%

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

7. Simplified 83.7%

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

8. Taylor expanded in beta around 0 69.3%

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

   1. associate-*r/ 69.3%

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

   2. metadata-eval 69.3%

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

   3. +-commutative 69.3%

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

   4. +-commutative 69.3%

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

10. Simplified 69.3%

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

11. Taylor expanded in alpha around 0 49.9%

    \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \alpha} \]
12. Step-by-step derivation

    1. *-commutative 49.9%

       \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776} \]

13. Simplified 49.9%

    \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.027777777777777776} \]
14. Add Preprocessing

Alternative 16: 44.8% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Simplified 85.3%

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

5. Taylor expanded in alpha around -inf 83.7%

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

   1. mul-1-neg 83.7%

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

   2. distribute-rgt-neg-in 83.7%

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

   3. sub-neg 83.7%

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

   4. associate-*r/ 83.7%

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

   5. distribute-lft-in 83.7%

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

   6. metadata-eval 83.7%

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

   7. mul-1-neg 83.7%

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

   8. unsub-neg 83.7%

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

   9. metadata-eval 83.7%

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

7. Simplified 83.7%

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

8. Taylor expanded in beta around 0 69.3%

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

   1. associate-*r/ 69.3%

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

   2. metadata-eval 69.3%

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

   3. +-commutative 69.3%

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

   4. +-commutative 69.3%

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

10. Simplified 69.3%

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

11. Taylor expanded in alpha around 0 50.0%

    \[\leadsto \color{blue}{0.08333333333333333} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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)))