Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 14.7s

Alternatives: 21

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: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Simplified 88.9%

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

   1. times-frac 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

   3. +-commutative 97.4%

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

   4. +-commutative 97.4%

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

   5. +-commutative 97.4%

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

   6. +-commutative 97.4%

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

   7. +-commutative 97.4%

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

   8. +-commutative 97.4%

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

   9. +-commutative 97.4%

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

   10. +-commutative 97.4%

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

   11. +-commutative 97.4%

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

8. Simplified 97.4%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. div-inv 99.8%

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

12. Applied egg-rr 99.8%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{t\_0}}{t\_0} \cdot \frac{1 + \beta}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 4.5e+15)
     (/ 1.0 (* t_0 (* (+ beta 2.0) (/ (+ beta 3.0) (+ 1.0 beta)))))
     (*
      (/ (* (+ 1.0 alpha) (/ 1.0 t_0)) t_0)
      (/ (+ 1.0 beta) (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.5e+15) {
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = (((1.0 + alpha) * (1.0 / t_0)) / t_0) * ((1.0 + beta) / (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 4.5d+15) then
        tmp = 1.0d0 / (t_0 * ((beta + 2.0d0) * ((beta + 3.0d0) / (1.0d0 + beta))))
    else
        tmp = (((1.0d0 + alpha) * (1.0d0 / t_0)) / t_0) * ((1.0d0 + beta) / (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.5e+15) {
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = (((1.0 + alpha) * (1.0 / t_0)) / t_0) * ((1.0 + beta) / (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 4.5e+15:
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))))
	else:
		tmp = (((1.0 + alpha) * (1.0 / t_0)) / t_0) * ((1.0 + beta) / (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 4.5e+15)
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(Float64(beta + 3.0) / Float64(1.0 + beta)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / t_0)) / t_0) * Float64(Float64(1.0 + beta) / Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 4.5e+15)
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	else
		tmp = (((1.0 + alpha) * (1.0 / t_0)) / t_0) * ((1.0 + beta) / (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.5e+15], N[(1.0 / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5e15

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. metadata-eval 99.8%

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

   3. Simplified 96.7%

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

      1. associate-+r+ 96.7%

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

      2. fma-undefine 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-+l+ 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+l+ 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*r* 96.7%

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

      9. associate-+r+ 96.7%

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

      10. +-commutative 96.7%

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

      11. associate-/l/ 99.8%

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

      12. clear-num 99.8%

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

      13. inv-pow 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in alpha around 0 63.7%

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

       1. associate-/l* 63.7%

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

       2. +-commutative 63.7%

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

       3. +-commutative 63.7%

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

   11. Simplified 63.7%

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

   if 4.5e15 < beta

   1. Initial program 87.8%

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

   3. Simplified 72.3%

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

      1. times-frac 92.3%

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

      2. +-commutative 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. associate-*r/ 92.3%

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

      2. +-commutative 92.3%

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

      3. +-commutative 92.3%

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

      4. +-commutative 92.3%

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

      5. +-commutative 92.3%

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

      6. +-commutative 92.3%

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

      7. +-commutative 92.3%

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

      8. +-commutative 92.3%

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

      9. +-commutative 92.3%

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

      10. +-commutative 92.3%

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

      11. +-commutative 92.3%

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

   8. Simplified 92.3%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. div-inv 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in beta around inf 99.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\alpha + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.25e+16)
     (/ 1.0 (* t_0 (* (+ beta 2.0) (/ (+ beta 3.0) (+ 1.0 beta)))))
     (*
      (/ (+ 1.0 beta) (+ alpha (+ beta 3.0)))
      (/ (/ (+ 1.0 alpha) beta) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.25e+16) {
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 + beta) / (alpha + (beta + 3.0))) * (((1.0 + alpha) / beta) / t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.25d+16) then
        tmp = 1.0d0 / (t_0 * ((beta + 2.0d0) * ((beta + 3.0d0) / (1.0d0 + beta))))
    else
        tmp = ((1.0d0 + beta) / (alpha + (beta + 3.0d0))) * (((1.0d0 + alpha) / beta) / t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.25e+16) {
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 + beta) / (alpha + (beta + 3.0))) * (((1.0 + alpha) / beta) / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.25e+16:
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))))
	else:
		tmp = ((1.0 + beta) / (alpha + (beta + 3.0))) * (((1.0 + alpha) / beta) / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.25e+16)
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(Float64(beta + 3.0) / Float64(1.0 + beta)))));
	else
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(Float64(1.0 + alpha) / beta) / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.25e+16)
		tmp = 1.0 / (t_0 * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	else
		tmp = ((1.0 + beta) / (alpha + (beta + 3.0))) * (((1.0 + alpha) / beta) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.25e+16], N[(1.0 / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25e16

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. metadata-eval 99.8%

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

   3. Simplified 96.7%

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

      1. associate-+r+ 96.7%

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

      2. fma-undefine 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-+l+ 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+l+ 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*r* 96.7%

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

      9. associate-+r+ 96.7%

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

      10. +-commutative 96.7%

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

      11. associate-/l/ 99.8%

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

      12. clear-num 99.8%

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

      13. inv-pow 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in alpha around 0 63.7%

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

       1. associate-/l* 63.7%

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

       2. +-commutative 63.7%

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

       3. +-commutative 63.7%

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

   11. Simplified 63.7%

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

   if 1.25e16 < beta

   1. Initial program 87.8%

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

   3. Simplified 72.3%

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

      1. times-frac 92.3%

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

      2. +-commutative 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. associate-*r/ 92.3%

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

      2. +-commutative 92.3%

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

      3. +-commutative 92.3%

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

      4. +-commutative 92.3%

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

      5. +-commutative 92.3%

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

      6. +-commutative 92.3%

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

      7. +-commutative 92.3%

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

      8. +-commutative 92.3%

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

      9. +-commutative 92.3%

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

      10. +-commutative 92.3%

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

      11. +-commutative 92.3%

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

   8. Simplified 92.3%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in beta around inf 89.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Simplified 88.9%

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

   1. times-frac 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

   3. +-commutative 97.4%

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

   4. +-commutative 97.4%

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

   5. +-commutative 97.4%

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

   6. +-commutative 97.4%

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

   7. +-commutative 97.4%

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

   8. +-commutative 97.4%

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

   9. +-commutative 97.4%

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

   10. +-commutative 97.4%

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

   11. +-commutative 97.4%

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

8. Simplified 97.4%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 99.7% accurate, 1.4× speedup

Math

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

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 (+ alpha beta))))
   (* (/ (+ 1.0 alpha) t_0) (/ (/ (+ 1.0 beta) (+ 3.0 (+ alpha beta))) t_0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Simplified 88.9%

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

   1. times-frac 97.4%

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

   2. +-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

   3. +-commutative 97.4%

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

   4. +-commutative 97.4%

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

   5. +-commutative 97.4%

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

   6. +-commutative 97.4%

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

   7. +-commutative 97.4%

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

   8. +-commutative 97.4%

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

   9. +-commutative 97.4%

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

   10. +-commutative 97.4%

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

   11. +-commutative 97.4%

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

8. Simplified 97.4%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

12. Applied egg-rr 99.8%

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

    1. associate-/l* 99.8%

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

    2. associate-+r+ 99.8%

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

    3. +-commutative 99.8%

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

    4. +-commutative 99.8%

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

    5. associate-+r+ 99.8%

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

    6. +-commutative 99.8%

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

    7. +-commutative 99.8%

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

    8. associate-+r+ 99.8%

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

    9. +-commutative 99.8%

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

    10. +-commutative 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 6: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.42e+17)
   (/
    1.0
    (* (+ alpha (+ beta 2.0)) (* (+ beta 2.0) (/ (+ beta 3.0) (+ 1.0 beta)))))
   (/ (+ (/ 1.0 beta) (/ alpha beta)) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.42e+17) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.42d+17) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * ((beta + 2.0d0) * ((beta + 3.0d0) / (1.0d0 + beta))))
    else
        tmp = ((1.0d0 / beta) + (alpha / beta)) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.42e+17) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.42e+17:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))))
	else:
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.42e+17)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) * Float64(Float64(beta + 3.0) / Float64(1.0 + beta)))));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.42e+17)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((beta + 2.0) * ((beta + 3.0) / (1.0 + beta))));
	else
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.42e+17], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.42e17

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. metadata-eval 99.8%

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

   3. Simplified 96.7%

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

      1. associate-+r+ 96.7%

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

      2. fma-undefine 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-+l+ 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+l+ 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*r* 96.7%

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

      9. associate-+r+ 96.7%

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

      10. +-commutative 96.7%

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

      11. associate-/l/ 99.8%

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

      12. clear-num 99.8%

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

      13. inv-pow 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in alpha around 0 63.7%

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

       1. associate-/l* 63.7%

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

       2. +-commutative 63.7%

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

       3. +-commutative 63.7%

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

   11. Simplified 63.7%

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

   if 1.42e17 < beta

   1. Initial program 87.8%

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

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + 3}{1 + \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.35e+17)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (+ (/ 1.0 beta) (/ alpha beta)) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35e+17) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.35d+17) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 / beta) + (alpha / beta)) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35e+17) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.35e+17:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.35e+17)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.35e+17)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 / beta) + (alpha / beta)) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.35e+17], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.35e17

   1. Initial program 99.8%

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

   3. Simplified 96.7%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 83.8%

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

       1. +-commutative 83.8%

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

   11. Simplified 83.8%

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

   12. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. *-commutative 62.7%

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

       3. +-commutative 62.7%

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

   14. Simplified 62.7%

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

   if 1.35e17 < beta

   1. Initial program 87.8%

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

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (* (+ 1.0 alpha) (/ 1.0 beta)) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9e15

   1. Initial program 99.8%

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

   3. Simplified 96.7%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 83.8%

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

       1. +-commutative 83.8%

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

   11. Simplified 83.8%

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

   12. Taylor expanded in alpha around 0 62.7%

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

       1. +-commutative 62.7%

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

       2. *-commutative 62.7%

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

       3. +-commutative 62.7%

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

   14. Simplified 62.7%

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

   if 9e15 < beta

   1. Initial program 87.8%

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

      1. clear-num 89.5%

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

      2. inv-pow 89.5%

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

   5. Applied egg-rr 89.5%

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

      1. unpow-1 89.5%

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

   7. Simplified 89.5%

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

      1. div-inv 89.4%

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

      2. metadata-eval 89.4%

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

      3. associate-+l+ 89.4%

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

      4. metadata-eval 89.4%

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

      5. associate-+r+ 89.4%

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

      6. associate-/r/ 89.4%

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

   9. Applied egg-rr 89.4%

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

       1. associate-*r/ 89.5%

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

       2. *-commutative 89.5%

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

       3. *-lft-identity 89.5%

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

       4. *-commutative 89.5%

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

       5. associate-+r+ 89.5%

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

       6. +-commutative 89.5%

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

       7. +-commutative 89.5%

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

   11. Simplified 89.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.0d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / beta)) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.0], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.8%

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

   3. Simplified 96.6%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 84.1%

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

       1. +-commutative 84.1%

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

   11. Simplified 84.1%

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

   12. Taylor expanded in beta around 0 83.5%

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

   if 3 < beta

   1. Initial program 88.2%

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

   3. Taylor expanded in beta around inf 87.5%

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

      1. clear-num 87.5%

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

      2. inv-pow 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. unpow-1 87.5%

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

   7. Simplified 87.5%

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

      1. div-inv 87.5%

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

      2. metadata-eval 87.5%

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

      3. associate-+l+ 87.5%

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

      4. metadata-eval 87.5%

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

      5. associate-+r+ 87.5%

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

      6. associate-/r/ 87.5%

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

   9. Applied egg-rr 87.5%

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

       1. associate-*r/ 87.5%

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

       2. *-commutative 87.5%

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

       3. *-lft-identity 87.5%

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

       4. *-commutative 87.5%

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

       5. associate-+r+ 87.5%

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

       6. +-commutative 87.5%

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

       7. +-commutative 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 84.9%

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

Alternative 10: 93.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{1 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (if (<= beta 1.35e+154)
     (/ 1.0 (* beta (+ beta 3.0)))
     (/ (/ alpha beta) (+ 1.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else if (beta <= 1.35e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (1.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else if (beta <= 1.35d+154) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / (1.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else if (beta <= 1.35e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (1.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	elif beta <= 1.35e+154:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / (1.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	elseif (beta <= 1.35e+154)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(1.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	elseif (beta <= 1.35e+154)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / (1.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 96.6%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 84.1%

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

       1. +-commutative 84.1%

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

   11. Simplified 84.1%

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

   12. Taylor expanded in beta around 0 83.5%

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

   if 2.2999999999999998 < beta < 1.35000000000000003e154

   1. Initial program 94.2%

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

   3. Taylor expanded in beta around inf 74.1%

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

   4. Taylor expanded in alpha around 0 66.9%

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

      1. +-commutative 66.9%

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

   6. Simplified 66.9%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 83.6%

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

   3. Taylor expanded in beta around inf 97.9%

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

   4. Taylor expanded in alpha around inf 97.2%

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

   5. Taylor expanded in beta around inf 97.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{1 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (+ (/ 1.0 beta) (/ alpha beta)) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 / beta) + (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 <= 2.3d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 / beta) + (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 <= 2.3) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 / beta) + (alpha / beta)) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) + Float64(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 <= 2.3)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 / beta) + (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, 2.3], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 96.6%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 84.1%

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

       1. +-commutative 84.1%

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

   11. Simplified 84.1%

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

   12. Taylor expanded in beta around 0 83.5%

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

   if 2.2999999999999998 < beta

   1. Initial program 88.2%

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

   3. Taylor expanded in beta around inf 87.5%

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

      1. clear-num 87.5%

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

      2. inv-pow 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. unpow-1 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in alpha around 0 87.4%

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

      1. +-commutative 87.4%

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

   10. Simplified 87.4%

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

   11. Taylor expanded in alpha around 0 87.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} 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 <= 3.6d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    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 <= 3.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} 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 <= 3.6:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	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 <= 3.6)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	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 <= 3.6)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	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, 3.6], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $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 < 3.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 96.6%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 84.1%

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

       1. +-commutative 84.1%

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

   11. Simplified 84.1%

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

   12. Taylor expanded in beta around 0 83.5%

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

   if 3.60000000000000009 < beta

   1. Initial program 88.2%

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

   3. Taylor expanded in beta around inf 87.5%

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

   4. Taylor expanded in alpha around 0 87.4%

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

      1. +-commutative 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\beta}{1 + \alpha}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ 1.0 (/ beta (+ 1.0 alpha))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / (beta / (1.0 + alpha))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.8d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = (1.0d0 / (beta / (1.0d0 + alpha))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / (beta / (1.0 + alpha))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = (1.0 / (beta / (1.0 + alpha))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(beta / Float64(1.0 + alpha))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = (1.0 / (beta / (1.0 + alpha))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 99.8%

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

   3. Simplified 96.6%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 84.1%

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

       1. +-commutative 84.1%

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

   11. Simplified 84.1%

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

   12. Taylor expanded in beta around 0 83.5%

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

   if 3.7999999999999998 < beta

   1. Initial program 88.2%

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

   3. Taylor expanded in beta around inf 87.5%

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

      1. clear-num 87.5%

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

      2. inv-pow 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. unpow-1 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in alpha around 0 87.4%

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

      1. +-commutative 87.4%

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

   10. Simplified 87.4%

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

   11. Taylor expanded in beta around inf 87.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\beta}{1 + \alpha}}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 95.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ 1.0 (* beta (/ beta (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} 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 <= 3.9d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    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 <= 3.9) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} 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 <= 3.9:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	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 <= 3.9)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	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 <= 3.9)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	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, 3.9], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $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 < 3.89999999999999991

   1. Initial program 99.8%

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

   3. Simplified 96.6%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in alpha around 0 84.1%

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

       1. +-commutative 84.1%

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

   11. Simplified 84.1%

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

   12. Taylor expanded in beta around 0 83.5%

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

   if 3.89999999999999991 < beta

   1. Initial program 88.2%

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

   3. Taylor expanded in beta around inf 87.5%

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

      1. clear-num 86.9%

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

      2. inv-pow 86.9%

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

      3. metadata-eval 86.9%

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

      4. associate-+l+ 86.9%

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

      5. metadata-eval 86.9%

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

      6. associate-+r+ 86.9%

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

   5. Applied egg-rr 86.9%

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

      1. unpow-1 86.9%

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

      2. associate-/r/ 86.9%

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

      3. associate-+r+ 86.9%

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

      4. +-commutative 86.9%

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

      5. +-commutative 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in beta around inf 86.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 37.6%

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

      1. clear-num 37.3%

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

      2. inv-pow 37.3%

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

      3. metadata-eval 37.3%

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

      4. associate-+l+ 37.3%

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

      5. metadata-eval 37.3%

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

      6. associate-+r+ 37.3%

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

   5. Applied egg-rr 37.3%

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

      1. unpow-1 37.3%

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

      2. associate-/r/ 37.3%

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

      3. associate-+r+ 37.3%

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

      4. +-commutative 37.3%

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

      5. +-commutative 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in alpha around 0 36.8%

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

      1. associate-/r* 37.0%

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

      2. +-commutative 37.0%

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

   10. Simplified 37.0%

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

   if 1 < alpha

   1. Initial program 88.5%

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

   3. Taylor expanded in beta around inf 18.1%

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

      1. clear-num 18.1%

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

      2. inv-pow 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. unpow-1 18.1%

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

   7. Simplified 18.1%

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

   8. Taylor expanded in alpha around 0 17.8%

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

      1. +-commutative 17.8%

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

   10. Simplified 17.8%

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

   11. Taylor expanded in alpha around inf 17.7%

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

Alternative 16: 56.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf 37.6%

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

      1. clear-num 37.3%

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

      2. inv-pow 37.3%

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

      3. metadata-eval 37.3%

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

      4. associate-+l+ 37.3%

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

      5. metadata-eval 37.3%

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

      6. associate-+r+ 37.3%

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

   5. Applied egg-rr 37.3%

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

      1. unpow-1 37.3%

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

      2. associate-/r/ 37.3%

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

      3. associate-+r+ 37.3%

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

      4. +-commutative 37.3%

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

      5. +-commutative 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in alpha around 0 36.8%

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

      1. associate-/r* 37.0%

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

      2. +-commutative 37.0%

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

   10. Simplified 37.0%

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

   if 1.1000000000000001 < alpha

   1. Initial program 88.5%

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

   3. Taylor expanded in beta around inf 18.1%

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

   4. Taylor expanded in alpha around inf 18.0%

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

   5. Taylor expanded in beta around inf 17.7%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.1:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{1 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 37.6%

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

      1. clear-num 37.3%

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

      2. inv-pow 37.3%

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

      3. metadata-eval 37.3%

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

      4. associate-+l+ 37.3%

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

      5. metadata-eval 37.3%

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

      6. associate-+r+ 37.3%

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

   5. Applied egg-rr 37.3%

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

      1. unpow-1 37.3%

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

      2. associate-/r/ 37.3%

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

      3. associate-+r+ 37.3%

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

      4. +-commutative 37.3%

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

      5. +-commutative 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in alpha around 0 36.8%

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

      1. associate-/r* 37.0%

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

      2. +-commutative 37.0%

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

   10. Simplified 37.0%

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

   if 1 < alpha

   1. Initial program 88.5%

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

   3. Taylor expanded in beta around inf 18.1%

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

      1. clear-num 18.1%

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

      2. inv-pow 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. unpow-1 18.1%

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

   7. Simplified 18.1%

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

   8. Taylor expanded in alpha around 0 17.8%

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

      1. +-commutative 17.8%

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

   10. Simplified 17.8%

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

   11. Taylor expanded in alpha around inf 13.3%

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

       1. +-commutative 13.3%

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

   13. Simplified 13.3%

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

Alternative 18: 53.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 37.6%

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

   4. Taylor expanded in alpha around 0 36.8%

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

      1. +-commutative 36.8%

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

   6. Simplified 36.8%

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

   if 1 < alpha

   1. Initial program 88.5%

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

   3. Taylor expanded in beta around inf 18.1%

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

      1. clear-num 18.1%

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

      2. inv-pow 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. unpow-1 18.1%

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

   7. Simplified 18.1%

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

   8. Taylor expanded in alpha around 0 17.8%

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

      1. +-commutative 17.8%

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

   10. Simplified 17.8%

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

   11. Taylor expanded in alpha around inf 13.3%

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

       1. +-commutative 13.3%

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

   13. Simplified 13.3%

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

Alternative 19: 51.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in beta around inf 30.9%

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

4. Taylor expanded in alpha around 0 28.3%

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

   1. +-commutative 28.3%

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

6. Simplified 28.3%

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

Alternative 20: 51.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\beta \cdot \beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in beta around inf 30.9%

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

   1. clear-num 30.6%

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

   2. inv-pow 30.6%

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

   3. metadata-eval 30.6%

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

   4. associate-+l+ 30.6%

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

   5. metadata-eval 30.6%

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

   6. associate-+r+ 30.6%

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

5. Applied egg-rr 30.6%

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

   1. unpow-1 30.6%

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

   2. associate-/r/ 30.7%

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

   3. associate-+r+ 30.7%

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

   4. +-commutative 30.7%

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

   5. +-commutative 30.7%

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

7. Simplified 30.7%

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

8. Taylor expanded in alpha around 0 28.3%

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

   1. +-commutative 28.3%

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

10. Simplified 28.3%

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

11. Taylor expanded in beta around inf 28.7%

    \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
12. Add Preprocessing

Alternative 21: 6.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in beta around inf 30.9%

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

   1. clear-num 30.6%

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

   2. inv-pow 30.6%

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

   3. metadata-eval 30.6%

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

   4. associate-+l+ 30.6%

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

   5. metadata-eval 30.6%

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

   6. associate-+r+ 30.6%

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

5. Applied egg-rr 30.6%

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

   1. unpow-1 30.6%

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

   2. associate-/r/ 30.7%

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

   3. associate-+r+ 30.7%

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

   4. +-commutative 30.7%

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

   5. +-commutative 30.7%

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

7. Simplified 30.7%

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

8. Taylor expanded in alpha around 0 28.3%

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

   1. +-commutative 28.3%

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

10. Simplified 28.3%

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

11. Taylor expanded in beta around 0 4.2%

    \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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)))