Octave 3.8, jcobi/3

Percentage Accurate: 93.9% → 99.8%

Time: 15.9s

Alternatives: 15

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	return Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 + beta) / t_0)) / t_0) / 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 + beta) / t_0)) / t_0) / (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[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.5%

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

3. Simplified 96.9%

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

   1. clear-num 96.9%

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

   2. associate-+r+ 96.9%

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

   3. *-commutative 96.9%

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

   4. frac-times 91.9%

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

   5. *-un-lft-identity 91.9%

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

   6. +-commutative 91.9%

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

   7. *-commutative 91.9%

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

   8. associate-+r+ 91.9%

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

5. Applied egg-rr 91.9%

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

   1. associate-/r* 96.9%

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

   2. associate-/l* 93.8%

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

   3. associate-*l/ 96.9%

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

   4. *-commutative 96.9%

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

   5. times-frac 99.7%

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

   6. associate-/r* 96.9%

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

   7. *-commutative 96.9%

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

   8. associate-/r* 99.7%

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

   9. +-commutative 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

   12. +-commutative 99.7%

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

7. Simplified 99.7%

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

   1. frac-times 96.9%

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

   2. +-commutative 96.9%

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

   3. +-commutative 96.9%

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

   4. +-commutative 96.9%

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

   5. associate-+r+ 96.9%

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

   6. +-commutative 96.9%

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

   7. *-commutative 96.9%

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

   8. associate-/r* 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \alpha}{t_1} \cdot \frac{1 + \beta}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 - \frac{\alpha}{\beta}\right)}{\left(\alpha + \beta\right) + 2}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.3e+146)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_0 t_1)))
     (/
      (/ (* (+ 1.0 alpha) (- 1.0 (/ alpha beta))) (+ (+ alpha beta) 2.0))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.3e+146) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) * (1.0 - (alpha / beta))) / ((alpha + beta) + 2.0)) / 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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 3.3d+146) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_0 * t_1))
    else
        tmp = (((1.0d0 + alpha) * (1.0d0 - (alpha / beta))) / ((alpha + beta) + 2.0d0)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.3e+146) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) * (1.0 - (alpha / beta))) / ((alpha + beta) + 2.0)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.3e+146:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1))
	else:
		tmp = (((1.0 + alpha) * (1.0 - (alpha / beta))) / ((alpha + beta) + 2.0)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.3e+146)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 - Float64(alpha / beta))) / Float64(Float64(alpha + beta) + 2.0)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.3e+146)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	else
		tmp = (((1.0 + alpha) * (1.0 - (alpha / beta))) / ((alpha + beta) + 2.0)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.3e+146], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 - N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.30000000000000016e146

   1. Initial program 97.9%

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

   3. Simplified 98.1%

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

   if 3.30000000000000016e146 < beta

   1. Initial program 83.2%

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

   3. Simplified 90.5%

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

      1. clear-num 90.5%

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

      2. associate-+r+ 90.5%

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

      3. *-commutative 90.5%

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

      4. frac-times 84.1%

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

      5. *-un-lft-identity 84.1%

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

      6. +-commutative 84.1%

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

      7. *-commutative 84.1%

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

      8. associate-+r+ 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. associate-/r* 90.5%

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

      2. associate-/l* 78.6%

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

      3. associate-*l/ 90.5%

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

      4. *-commutative 90.5%

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

      5. times-frac 99.9%

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

      6. associate-/r* 90.5%

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

      7. *-commutative 90.5%

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

      8. associate-/r* 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. frac-times 90.5%

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

      2. +-commutative 90.5%

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

      3. +-commutative 90.5%

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

      4. +-commutative 90.5%

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

      5. associate-+r+ 90.5%

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

      6. +-commutative 90.5%

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

      7. *-commutative 90.5%

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

      8. associate-/r* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in beta around inf 95.0%

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

       1. mul-1-neg 95.0%

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

       2. unsub-neg 95.0%

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

   12. Simplified 95.0%

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

   13. Taylor expanded in alpha around inf 95.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 - \frac{\alpha}{\beta}\right)}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Simplified 96.9%

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

   1. clear-num 96.9%

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

   2. associate-+r+ 96.9%

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

   3. *-commutative 96.9%

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

   4. frac-times 91.9%

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

   5. *-un-lft-identity 91.9%

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

   6. +-commutative 91.9%

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

   7. *-commutative 91.9%

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

   8. associate-+r+ 91.9%

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

5. Applied egg-rr 91.9%

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

   1. associate-/r* 96.9%

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

   2. associate-/l* 93.8%

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

   3. associate-*l/ 96.9%

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

   4. *-commutative 96.9%

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

   5. times-frac 99.7%

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

   6. associate-/r* 96.9%

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

   7. *-commutative 96.9%

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

   8. associate-/r* 99.7%

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

   9. +-commutative 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

   12. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.3e15

   1. Initial program 99.7%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in alpha around 0 75.7%

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

   5. Taylor expanded in alpha around 0 60.3%

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

   if 2.3e15 < beta

   1. Initial program 85.6%

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

   3. Simplified 90.9%

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

      1. clear-num 90.9%

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

      2. associate-+r+ 90.9%

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

      3. *-commutative 90.9%

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

      4. frac-times 74.2%

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

      5. *-un-lft-identity 74.2%

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

      6. +-commutative 74.2%

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

      7. *-commutative 74.2%

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

      8. associate-+r+ 74.2%

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

   5. Applied egg-rr 74.2%

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

      1. associate-/r* 91.1%

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

      2. associate-/l* 80.5%

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

      3. associate-*l/ 91.2%

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

      4. *-commutative 91.2%

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

      5. times-frac 99.6%

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

      6. associate-/r* 90.9%

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

      7. *-commutative 90.9%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. frac-times 91.2%

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

      2. +-commutative 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+r+ 91.2%

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

      6. +-commutative 91.2%

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

      7. *-commutative 91.2%

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

      8. associate-/r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in beta around inf 81.2%

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

       1. mul-1-neg 81.2%

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

       2. unsub-neg 81.2%

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

   12. Simplified 81.2%

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

   13. Taylor expanded in alpha around inf 81.2%

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

4. Final simplification 66.5%

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

Alternative 5: 97.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.2d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.2)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.2], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2000000000000002

   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 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-+r+ 99.4%

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

      3. associate-/r* 99.8%

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

      4. frac-times 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-+r+ 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-in 99.4%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in beta around 0 90.8%

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

      1. associate-/r* 97.3%

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

      2. +-commutative 97.3%

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

      3. distribute-rgt-in 97.3%

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

      4. +-commutative 97.3%

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

      5. +-commutative 97.3%

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

   10. Simplified 97.3%

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

   if 3.2000000000000002 < beta

   1. Initial program 87.2%

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

   3. Simplified 91.9%

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

      1. clear-num 91.9%

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

      2. associate-+r+ 91.9%

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

      3. *-commutative 91.9%

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

      4. frac-times 77.2%

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

      5. *-un-lft-identity 77.2%

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

      6. +-commutative 77.2%

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

      7. *-commutative 77.2%

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

      8. associate-+r+ 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. associate-/r* 92.1%

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

      2. associate-/l* 82.7%

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

      3. associate-*l/ 92.1%

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

      4. *-commutative 92.1%

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

      5. times-frac 99.6%

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

      6. associate-/r* 91.9%

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

      7. *-commutative 91.9%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. frac-times 92.1%

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

      2. +-commutative 92.1%

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

      3. +-commutative 92.1%

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

      4. +-commutative 92.1%

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

      5. associate-+r+ 92.1%

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

      6. +-commutative 92.1%

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

      7. *-commutative 92.1%

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

      8. associate-/r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in beta around inf 73.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6: 97.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 2.0)) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.3d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 2.0d0)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.3:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + beta) + 2.0)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.3], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.30000000000000004

   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 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-+r+ 99.4%

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

      3. associate-/r* 99.8%

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

      4. frac-times 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-+r+ 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-in 99.4%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in beta around 0 90.8%

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

      1. associate-/r* 97.3%

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

      2. +-commutative 97.3%

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

      3. distribute-rgt-in 97.3%

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

      4. +-commutative 97.3%

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

      5. +-commutative 97.3%

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

   10. Simplified 97.3%

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

   if 1.30000000000000004 < beta

   1. Initial program 87.2%

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

   3. Simplified 91.9%

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

      1. clear-num 91.9%

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

      2. associate-+r+ 91.9%

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

      3. *-commutative 91.9%

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

      4. frac-times 77.2%

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

      5. *-un-lft-identity 77.2%

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

      6. +-commutative 77.2%

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

      7. *-commutative 77.2%

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

      8. associate-+r+ 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. associate-/r* 92.1%

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

      2. associate-/l* 82.7%

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

      3. associate-*l/ 92.1%

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

      4. *-commutative 92.1%

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

      5. times-frac 99.6%

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

      6. associate-/r* 91.9%

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

      7. *-commutative 91.9%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. frac-times 92.1%

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

      2. +-commutative 92.1%

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

      3. +-commutative 92.1%

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

      4. +-commutative 92.1%

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

      5. associate-+r+ 92.1%

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

      6. +-commutative 92.1%

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

      7. *-commutative 92.1%

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

      8. associate-/r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in beta around inf 74.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7: 98.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ (+ alpha beta) 2.0)) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / ((alpha + beta) + 2.0d0)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2e+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(Float64(alpha + beta) + 2.0)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / ((alpha + beta) + 2.0)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2e+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[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2e15

   1. Initial program 99.7%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in alpha around 0 75.7%

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

   5. Taylor expanded in alpha around 0 60.3%

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

   if 5.2e15 < beta

   1. Initial program 85.6%

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

   3. Simplified 90.9%

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

      1. clear-num 90.9%

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

      2. associate-+r+ 90.9%

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

      3. *-commutative 90.9%

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

      4. frac-times 74.2%

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

      5. *-un-lft-identity 74.2%

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

      6. +-commutative 74.2%

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

      7. *-commutative 74.2%

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

      8. associate-+r+ 74.2%

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

   5. Applied egg-rr 74.2%

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

      1. associate-/r* 91.1%

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

      2. associate-/l* 80.5%

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

      3. associate-*l/ 91.2%

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

      4. *-commutative 91.2%

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

      5. times-frac 99.6%

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

      6. associate-/r* 90.9%

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

      7. *-commutative 90.9%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. frac-times 91.2%

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

      2. +-commutative 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+r+ 91.2%

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

      6. +-commutative 91.2%

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

      7. *-commutative 91.2%

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

      8. associate-/r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in beta around inf 81.0%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 11.6:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 11.6)
   (/ (/ 0.5 (+ beta 2.0)) (+ alpha (+ beta 3.0)))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 11.6) {
		tmp = (0.5 / (beta + 2.0)) / (alpha + (beta + 3.0));
	} else {
		tmp = ((1.0 + 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 <= 11.6d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (alpha + (beta + 3.0d0))
    else
        tmp = ((1.0d0 + 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 <= 11.6) {
		tmp = (0.5 / (beta + 2.0)) / (alpha + (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 11.6:
		tmp = (0.5 / (beta + 2.0)) / (alpha + (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 11.6)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + 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 <= 11.6)
		tmp = (0.5 / (beta + 2.0)) / (alpha + (beta + 3.0));
	else
		tmp = ((1.0 + 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, 11.6], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 11.5999999999999996

   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 99.4%

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

      1. clear-num 99.4%

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

      2. associate-+r+ 99.4%

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

      3. *-commutative 99.4%

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

      4. frac-times 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. +-commutative 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-+r+ 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. associate-/r* 99.4%

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

      2. associate-/l* 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-commutative 99.4%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. frac-times 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. associate-+r+ 99.4%

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

      6. +-commutative 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-/r* 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in beta around 0 97.6%

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

   11. Taylor expanded in alpha around 0 60.0%

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

   if 11.5999999999999996 < beta

   1. Initial program 87.0%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in beta around inf 74.5%

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

   5. Taylor expanded in beta around inf 74.2%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 11.6:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 9: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \end{array} \]

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 3.0))))
   (if (<= beta 4.5)
     (/ (/ 0.5 (+ beta 2.0)) t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / t_0;
	} else {
		tmp = ((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 + 3.0d0)
    if (beta <= 4.5d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / t_0
    else
        tmp = ((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 + 3.0);
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (beta + 2.0)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 4.5:
		tmp = (0.5 / (beta + 2.0)) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / t_0);
	else
		tmp = 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 + 3.0);
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = (0.5 / (beta + 2.0)) / t_0;
	else
		tmp = ((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 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.5], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-+r+ 99.4%

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

      3. *-commutative 99.4%

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

      4. frac-times 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. +-commutative 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-+r+ 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. associate-/r* 99.4%

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

      2. associate-/l* 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-commutative 99.4%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. frac-times 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. associate-+r+ 99.4%

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

      6. +-commutative 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-/r* 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in beta around 0 97.6%

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

   11. Taylor expanded in alpha around 0 60.0%

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

   if 4.5 < beta

   1. Initial program 87.0%

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

   3. Simplified 91.8%

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

      1. clear-num 91.8%

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

      2. associate-+r+ 91.8%

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

      3. *-commutative 91.8%

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

      4. frac-times 76.9%

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

      5. *-un-lft-identity 76.9%

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

      6. +-commutative 76.9%

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

      7. *-commutative 76.9%

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

      8. associate-+r+ 76.9%

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

   5. Applied egg-rr 76.9%

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

      1. associate-/r* 92.0%

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

      2. associate-/l* 82.5%

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

      3. associate-*l/ 92.0%

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

      4. *-commutative 92.0%

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

      5. times-frac 99.6%

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

      6. associate-/r* 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. frac-times 92.0%

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

      2. +-commutative 92.0%

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

      3. +-commutative 92.0%

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

      4. +-commutative 92.0%

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

      5. associate-+r+ 92.0%

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

      6. +-commutative 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in beta around inf 74.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10: 96.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6)
   0.08333333333333333
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + 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 <= 3.6d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + 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 <= 3.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + 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 <= 3.6)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + 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, 3.6], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   1. Initial program 99.8%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in alpha around 0 76.5%

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

   5. Taylor expanded in beta around 0 74.4%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 74.4%

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

      2. +-commutative 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in alpha around 0 58.3%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 3.60000000000000009 < beta

   1. Initial program 87.0%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in beta around inf 74.5%

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

   5. Taylor expanded in beta around inf 74.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 11: 90.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6) 0.08333333333333333 (/ 1.0 (* beta (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / (beta * (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], 0.08333333333333333, N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in alpha around 0 76.4%

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

   5. Taylor expanded in beta around 0 74.3%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 74.3%

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

      2. +-commutative 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in alpha around 0 58.6%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.60000000000000009 < beta

   1. Initial program 87.2%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in beta around inf 73.7%

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

   5. Taylor expanded in alpha around 0 70.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 12: 90.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6) 0.08333333333333333 (/ (/ 1.0 (+ beta 2.0)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 / (beta + 2.0)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = (1.0d0 / (beta + 2.0d0)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 / (beta + 2.0)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = 0.08333333333333333
	else:
		tmp = (1.0 / (beta + 2.0)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = (1.0 / (beta + 2.0)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], 0.08333333333333333, N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in alpha around 0 76.4%

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

   5. Taylor expanded in beta around 0 74.3%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 74.3%

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

      2. +-commutative 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in alpha around 0 58.6%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.60000000000000009 < beta

   1. Initial program 87.2%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in beta around inf 73.7%

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

      1. un-div-inv 73.9%

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

      2. +-commutative 73.9%

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

      3. associate-+r+ 73.9%

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

      4. +-commutative 73.9%

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

   6. Applied egg-rr 73.9%

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

   7. Taylor expanded in alpha around 0 71.2%

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

      1. +-commutative 71.2%

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

   9. Simplified 71.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{\beta}\\ \end{array} \]

Alternative 13: 44.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{6 + \alpha \cdot 5} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.5 (+ 6.0 (* alpha 5.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / (6.0 + (alpha * 5.0));
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0 / (6.0d0 + (alpha * 5.0d0))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / (6.0 + (alpha * 5.0));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / (6.0 + (alpha * 5.0))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / Float64(6.0 + Float64(alpha * 5.0)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / (6.0 + (alpha * 5.0));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.5 / N[(6.0 + N[(alpha * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

   1. associate-/l/ 93.8%

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

   2. +-commutative 93.8%

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

   3. +-commutative 93.8%

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

   4. associate-+r+ 93.8%

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

   5. *-commutative 93.8%

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

   6. metadata-eval 93.8%

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

   7. associate-+l+ 93.8%

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

   8. metadata-eval 93.8%

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

   9. associate-+l+ 93.8%

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

   10. metadata-eval 93.8%

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

   11. metadata-eval 93.8%

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

   12. associate-+l+ 93.8%

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

3. Simplified 93.8%

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

4. Taylor expanded in alpha around 0 77.7%

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

5. Taylor expanded in alpha around 0 67.2%

   \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\alpha \cdot \left(5 + 2 \cdot \beta\right) + \left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

6. Taylor expanded in beta around 0 41.8%

   \[\leadsto \color{blue}{\frac{0.5}{6 + 5 \cdot \alpha}} \]
7. Step-by-step derivation

   1. *-commutative 41.8%

      \[\leadsto \frac{0.5}{6 + \color{blue}{\alpha \cdot 5}} \]

8. Simplified 41.8%

   \[\leadsto \color{blue}{\frac{0.5}{6 + \alpha \cdot 5}} \]

9. Final simplification 41.8%

   \[\leadsto \frac{0.5}{6 + \alpha \cdot 5} \]

Alternative 14: 45.9% accurate, 6.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   1. Initial program 99.8%

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

      1. associate-/l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. *-commutative 99.4%

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

      6. metadata-eval 99.4%

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

      7. associate-+l+ 99.4%

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

      8. metadata-eval 99.4%

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

      9. associate-+l+ 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in alpha around 0 76.5%

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

   5. Taylor expanded in beta around 0 74.4%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 74.4%

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

      2. +-commutative 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in alpha around 0 58.3%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 12 < beta

   1. Initial program 87.0%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in beta around inf 74.5%

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

      1. un-div-inv 74.7%

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

      2. +-commutative 74.7%

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

      3. associate-+r+ 74.7%

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

      4. +-commutative 74.7%

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

   6. Applied egg-rr 74.7%

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

   7. Taylor expanded in alpha around inf 7.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 15: 44.2% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

   1. associate-/l/ 93.8%

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

   2. +-commutative 93.8%

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

   3. +-commutative 93.8%

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

   4. associate-+r+ 93.8%

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

   5. *-commutative 93.8%

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

   6. metadata-eval 93.8%

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

   7. associate-+l+ 93.8%

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

   8. metadata-eval 93.8%

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

   9. associate-+l+ 93.8%

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

   10. metadata-eval 93.8%

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

   11. metadata-eval 93.8%

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

   12. associate-+l+ 93.8%

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

3. Simplified 93.8%

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

4. Taylor expanded in alpha around 0 77.7%

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

5. Taylor expanded in beta around 0 54.0%

   \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
6. Step-by-step derivation

   1. +-commutative 54.0%

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

   2. +-commutative 54.0%

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

7. Simplified 54.0%

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

8. Taylor expanded in alpha around 0 40.4%

   \[\leadsto \color{blue}{0.08333333333333333} \]

9. Final simplification 40.4%

   \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023321 
(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)))