Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 17.1s

Alternatives: 16

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	return ((beta + 1.0) / t_0) * (((1.0 + alpha) / 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 = (beta + alpha) + 2.0d0
    code = ((beta + 1.0d0) / t_0) * (((1.0d0 + alpha) / t_0) / (alpha + (beta + 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = ((beta + 1.0) / t_0) * (((1.0 + alpha) / 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.5%

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

   1. associate-/l/ 93.0%

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

   2. associate-+l+ 93.0%

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

   3. +-commutative 93.0%

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

   4. associate-+r+ 93.0%

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

   5. associate-+l+ 93.0%

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

   6. distribute-rgt1-in 93.0%

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

   7. *-rgt-identity 93.0%

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

   8. distribute-lft-out 93.0%

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

   9. +-commutative 93.0%

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

   10. associate-*l/ 95.4%

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

   11. *-commutative 95.4%

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

   12. associate-*r/ 91.8%

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

3. Simplified 91.8%

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

   1. associate-*r/ 95.4%

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

   2. +-commutative 95.4%

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

5. Applied egg-rr 95.4%

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

   1. +-commutative 95.4%

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

   2. *-commutative 95.4%

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

   3. +-commutative 95.4%

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

   4. associate-*r/ 95.4%

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

   5. associate-/r* 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-+r+ 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+140}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{t_1}}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{t_0} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+140)
     (* (+ 1.0 alpha) (/ (/ (+ beta 1.0) t_1) (* t_0 t_1)))
     (*
      (/ (/ (+ 1.0 alpha) (+ (+ beta alpha) 2.0)) t_0)
      (+ 1.0 (/ (- -1.0 alpha) beta))))))

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 <= 1e+140) {
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_1) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / t_0) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 1d+140) then
        tmp = (1.0d0 + alpha) * (((beta + 1.0d0) / t_1) / (t_0 * t_1))
    else
        tmp = (((1.0d0 + alpha) / ((beta + alpha) + 2.0d0)) / t_0) * (1.0d0 + (((-1.0d0) - alpha) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+140) {
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_1) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / t_0) * (1.0 + ((-1.0 - alpha) / beta));
	}
	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 <= 1e+140:
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_1) / (t_0 * t_1))
	else:
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / t_0) * (1.0 + ((-1.0 - alpha) / beta))
	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 <= 1e+140)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(beta + 1.0) / t_1) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(beta + alpha) + 2.0)) / t_0) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)));
	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 <= 1e+140)
		tmp = (1.0 + alpha) * (((beta + 1.0) / t_1) / (t_0 * t_1));
	else
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / t_0) * (1.0 + ((-1.0 - alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 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, 1e+140], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000006e140

   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/ 98.4%

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

      2. associate-+l+ 98.4%

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

      3. +-commutative 98.4%

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

      4. associate-+r+ 98.4%

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

      5. associate-+l+ 98.4%

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

      6. distribute-rgt1-in 98.4%

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

      7. *-rgt-identity 98.4%

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

      8. distribute-lft-out 98.4%

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

      9. +-commutative 98.4%

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

      10. associate-*l/ 98.4%

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

      11. *-commutative 98.4%

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

      12. associate-*r/ 94.0%

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

   3. Simplified 94.0%

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

   if 1.00000000000000006e140 < beta

   1. Initial program 71.3%

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

      1. associate-/l/ 69.6%

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

      2. associate-+l+ 69.6%

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

      3. +-commutative 69.6%

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

      4. associate-+r+ 69.6%

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

      5. associate-+l+ 69.6%

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

      6. distribute-rgt1-in 69.6%

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

      7. *-rgt-identity 69.6%

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

      8. distribute-lft-out 69.6%

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

      9. +-commutative 69.6%

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

      10. associate-*l/ 82.6%

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

      11. *-commutative 82.6%

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

      12. associate-*r/ 82.6%

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

   3. Simplified 82.6%

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

      1. associate-*r/ 82.6%

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

      2. +-commutative 82.6%

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

   5. Applied egg-rr 82.6%

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

      1. +-commutative 82.6%

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

      2. *-commutative 82.6%

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

      3. +-commutative 82.6%

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

      4. associate-*r/ 82.6%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in beta around inf 85.4%

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

      1. mul-1-neg 84.8%

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

      2. unsub-neg 84.8%

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

   10. Simplified 85.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+140}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 800000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 800000000.0)
   (/ (/ (+ beta 1.0) (* (+ beta 3.0) (+ beta 2.0))) (+ beta (+ alpha 2.0)))
   (*
    (/ (/ (+ 1.0 alpha) (+ (+ beta alpha) 2.0)) (+ alpha (+ beta 3.0)))
    (+ 1.0 (/ (- -1.0 alpha) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 800000000.0) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 800000000.0d0) then
        tmp = ((beta + 1.0d0) / ((beta + 3.0d0) * (beta + 2.0d0))) / (beta + (alpha + 2.0d0))
    else
        tmp = (((1.0d0 + alpha) / ((beta + alpha) + 2.0d0)) / (alpha + (beta + 3.0d0))) * (1.0d0 + (((-1.0d0) - alpha) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 800000000.0) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 800000000.0:
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0))
	else:
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 800000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))) / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(beta + alpha) + 2.0)) / Float64(alpha + Float64(beta + 3.0))) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 800000000.0)
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	else
		tmp = (((1.0 + alpha) / ((beta + alpha) + 2.0)) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 800000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8e8

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 93.9%

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

   3. Simplified 93.9%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 69.4%

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

       1. associate-/r* 69.4%

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

   12. Simplified 69.4%

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

       1. associate-*l/ 69.5%

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

       2. +-commutative 69.5%

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

       3. associate-/r* 69.5%

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

       4. associate-+l+ 69.5%

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

   14. Applied egg-rr 69.5%

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

       1. associate-*r/ 69.5%

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

       2. +-commutative 69.5%

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

       3. *-rgt-identity 69.5%

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

       4. +-commutative 69.5%

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

   16. Simplified 69.5%

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

   if 8e8 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ 80.1%

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

      2. associate-+l+ 80.1%

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

      3. +-commutative 80.1%

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

      4. associate-+r+ 80.1%

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

      5. associate-+l+ 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. *-rgt-identity 80.1%

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

      8. distribute-lft-out 80.1%

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

      9. +-commutative 80.1%

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

      10. associate-*l/ 87.6%

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

      11. *-commutative 87.6%

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

      12. associate-*r/ 87.6%

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

   3. Simplified 87.6%

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

      1. associate-*r/ 87.6%

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

      2. +-commutative 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. +-commutative 87.6%

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

      2. *-commutative 87.6%

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

      3. +-commutative 87.6%

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

      4. associate-*r/ 87.6%

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

      5. associate-/r* 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around inf 86.7%

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

      1. mul-1-neg 85.8%

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

      2. unsub-neg 85.8%

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

   10. Simplified 86.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 800000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8e+15)
   (/ (/ (+ beta 1.0) (* (+ beta 3.0) (+ beta 2.0))) (+ beta (+ alpha 2.0)))
   (*
    (+ 1.0 (/ (- -1.0 alpha) beta))
    (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8e+15) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((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 <= 7.8d+15) then
        tmp = ((beta + 1.0d0) / ((beta + 3.0d0) * (beta + 2.0d0))) / (beta + (alpha + 2.0d0))
    else
        tmp = (1.0d0 + (((-1.0d0) - alpha) / beta)) * (((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 <= 7.8e+15) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.8e+15:
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0))
	else:
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((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 <= 7.8e+15)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))) / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) * 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 <= 7.8e+15)
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	else
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((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, 7.8e+15], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 93.9%

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

   3. Simplified 93.9%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 69.4%

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

       1. associate-/r* 69.4%

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

   12. Simplified 69.4%

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

       1. associate-*l/ 69.5%

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

       2. +-commutative 69.5%

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

       3. associate-/r* 69.5%

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

       4. associate-+l+ 69.5%

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

   14. Applied egg-rr 69.5%

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

       1. associate-*r/ 69.5%

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

       2. +-commutative 69.5%

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

       3. *-rgt-identity 69.5%

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

       4. +-commutative 69.5%

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

   16. Simplified 69.5%

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

   if 7.8e15 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ 80.1%

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

      2. associate-+l+ 80.1%

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

      3. +-commutative 80.1%

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

      4. associate-+r+ 80.1%

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

      5. associate-+l+ 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. *-rgt-identity 80.1%

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

      8. distribute-lft-out 80.1%

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

      9. +-commutative 80.1%

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

      10. associate-*l/ 87.6%

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

      11. *-commutative 87.6%

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

      12. associate-*r/ 87.6%

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

   3. Simplified 87.6%

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

      1. associate-*r/ 87.6%

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

      2. +-commutative 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. +-commutative 87.6%

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

      2. *-commutative 87.6%

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

      3. +-commutative 87.6%

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

      4. associate-*r/ 87.6%

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

      5. associate-/r* 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around inf 86.4%

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

   9. Taylor expanded in beta around inf 85.8%

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

       1. mul-1-neg 85.8%

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

       2. unsub-neg 85.8%

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

   11. Simplified 85.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2e+15)
   (* (+ beta 1.0) (/ (/ 1.0 (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) (+ beta 3.0)) (+ beta 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+15) {
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (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 <= 8.2d+15) then
        tmp = (beta + 1.0d0) * ((1.0d0 / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + 3.0d0)) / (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 <= 8.2e+15) {
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2e+15:
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2e+15)
		tmp = Float64(Float64(beta + 1.0) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + 3.0)) / 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 <= 8.2e+15)
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (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, 8.2e+15], N[(N[(beta + 1.0), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*r/ 99.2%

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

      11. associate-*r/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in alpha around 0 83.8%

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

   5. Taylor expanded in alpha around 0 68.7%

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

   if 8.2e15 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ 80.1%

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

      2. associate-+l+ 80.1%

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

      3. +-commutative 80.1%

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

      4. associate-+r+ 80.1%

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

      5. associate-+l+ 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. *-rgt-identity 80.1%

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

      8. distribute-lft-out 80.1%

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

      9. +-commutative 80.1%

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

      10. associate-*l/ 87.6%

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

      11. *-commutative 87.6%

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

      12. associate-*r/ 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in alpha around 0 84.8%

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

   5. Taylor expanded in beta around inf 84.7%

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

      1. expm1-log1p-u 84.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-udef 46.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]

      3. un-div-inv 46.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 46.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 + \alpha}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1 \]

   7. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 84.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 84.7%

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

      3. associate-/r* 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \]

Alternative 6: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+16}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5e+16)
   (* (+ beta 1.0) (/ (/ 1.0 (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0))))
   (*
    (/ (+ beta 1.0) beta)
    (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5e+16) {
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (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 <= 6.5d+16) then
        tmp = (beta + 1.0d0) * ((1.0d0 / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((beta + 1.0d0) / beta) * (((1.0d0 + alpha) / (alpha + (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 <= 6.5e+16) {
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5e+16:
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5e+16)
		tmp = Float64(Float64(beta + 1.0) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(beta + 1.0) / beta) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 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 <= 6.5e+16)
		tmp = (beta + 1.0) * ((1.0 / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (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, 6.5e+16], N[(N[(beta + 1.0), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta + 1.0), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5e16

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*r/ 99.2%

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

      11. associate-*r/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in alpha around 0 83.8%

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

   5. Taylor expanded in alpha around 0 68.7%

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

   if 6.5e16 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ 80.1%

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

      2. associate-+l+ 80.1%

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

      3. +-commutative 80.1%

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

      4. associate-+r+ 80.1%

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

      5. associate-+l+ 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. *-rgt-identity 80.1%

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

      8. distribute-lft-out 80.1%

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

      9. +-commutative 80.1%

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

      10. associate-*r/ 87.6%

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

      11. associate-*r/ 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in beta around inf 68.7%

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

      1. unpow2 68.7%

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

   6. Simplified 68.7%

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

      1. associate-*r/ 84.7%

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

      2. +-commutative 84.7%

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

      3. +-commutative 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. times-frac 86.1%

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

      2. +-commutative 86.1%

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

   10. Simplified 86.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+16}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \]

Alternative 7: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.08e+16)
   (/ (/ (+ beta 1.0) (* (+ beta 3.0) (+ beta 2.0))) (+ beta (+ alpha 2.0)))
   (*
    (/ (+ beta 1.0) beta)
    (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.08e+16) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (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 <= 1.08d+16) then
        tmp = ((beta + 1.0d0) / ((beta + 3.0d0) * (beta + 2.0d0))) / (beta + (alpha + 2.0d0))
    else
        tmp = ((beta + 1.0d0) / beta) * (((1.0d0 + alpha) / (alpha + (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 <= 1.08e+16) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.08e+16:
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0))
	else:
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.08e+16)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))) / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(beta + 1.0) / beta) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 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 <= 1.08e+16)
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) / (beta + (alpha + 2.0));
	else
		tmp = ((beta + 1.0) / beta) * (((1.0 + alpha) / (alpha + (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, 1.08e+16], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta + 1.0), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.08e16

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 93.9%

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

   3. Simplified 93.9%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 69.4%

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

       1. associate-/r* 69.4%

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

   12. Simplified 69.4%

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

       1. associate-*l/ 69.5%

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

       2. +-commutative 69.5%

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

       3. associate-/r* 69.5%

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

       4. associate-+l+ 69.5%

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

   14. Applied egg-rr 69.5%

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

       1. associate-*r/ 69.5%

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

       2. +-commutative 69.5%

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

       3. *-rgt-identity 69.5%

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

       4. +-commutative 69.5%

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

   16. Simplified 69.5%

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

   if 1.08e16 < beta

   1. Initial program 83.2%

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

      1. associate-/l/ 80.1%

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

      2. associate-+l+ 80.1%

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

      3. +-commutative 80.1%

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

      4. associate-+r+ 80.1%

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

      5. associate-+l+ 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. *-rgt-identity 80.1%

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

      8. distribute-lft-out 80.1%

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

      9. +-commutative 80.1%

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

      10. associate-*r/ 87.6%

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

      11. associate-*r/ 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in beta around inf 68.7%

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

      1. unpow2 68.7%

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

   6. Simplified 68.7%

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

      1. associate-*r/ 84.7%

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

      2. +-commutative 84.7%

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

      3. +-commutative 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. times-frac 86.1%

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

      2. +-commutative 86.1%

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

   10. Simplified 86.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \]

Alternative 8: 97.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 0.8:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\beta \cdot -0.1388888888888889 + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 0.8)
   (*
    (/ (+ beta 1.0) (+ (+ beta alpha) 2.0))
    (+ (* beta -0.1388888888888889) 0.16666666666666666))
   (/ (/ (+ 1.0 alpha) (+ beta 3.0)) (+ beta 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.8) {
		tmp = ((beta + 1.0) / ((beta + alpha) + 2.0)) * ((beta * -0.1388888888888889) + 0.16666666666666666);
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (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 <= 0.8d0) then
        tmp = ((beta + 1.0d0) / ((beta + alpha) + 2.0d0)) * ((beta * (-0.1388888888888889d0)) + 0.16666666666666666d0)
    else
        tmp = ((1.0d0 + alpha) / (beta + 3.0d0)) / (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 <= 0.8) {
		tmp = ((beta + 1.0) / ((beta + alpha) + 2.0)) * ((beta * -0.1388888888888889) + 0.16666666666666666);
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 0.8:
		tmp = ((beta + 1.0) / ((beta + alpha) + 2.0)) * ((beta * -0.1388888888888889) + 0.16666666666666666)
	else:
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 0.8)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + alpha) + 2.0)) * Float64(Float64(beta * -0.1388888888888889) + 0.16666666666666666));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + 3.0)) / 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 <= 0.8)
		tmp = ((beta + 1.0) / ((beta + alpha) + 2.0)) * ((beta * -0.1388888888888889) + 0.16666666666666666);
	else
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (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, 0.8], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * -0.1388888888888889), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 0.80000000000000004

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 94.3%

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

   3. Simplified 94.3%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 70.2%

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

       1. associate-/r* 70.2%

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

   12. Simplified 70.2%

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

   13. Taylor expanded in beta around 0 69.3%

       \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\left(-0.1388888888888889 \cdot \beta + 0.16666666666666666\right)} \]

   if 0.80000000000000004 < beta

   1. Initial program 83.5%

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

      1. associate-/l/ 80.5%

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

      2. associate-+l+ 80.5%

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

      3. +-commutative 80.5%

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

      4. associate-+r+ 80.5%

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

      5. associate-+l+ 80.5%

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

      6. distribute-rgt1-in 80.6%

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

      7. *-rgt-identity 80.6%

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

      8. distribute-lft-out 80.6%

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

      9. +-commutative 80.6%

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

      10. associate-*l/ 87.9%

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

      11. *-commutative 87.9%

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

      12. associate-*r/ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in alpha around 0 82.8%

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

   5. Taylor expanded in beta around inf 82.8%

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

      1. expm1-log1p-u 82.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-udef 45.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]

      3. un-div-inv 45.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 45.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 + \alpha}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1 \]

   7. Applied egg-rr 45.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 82.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 82.8%

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

      3. associate-/r* 83.9%

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

   9. Simplified 83.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 0.8:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\beta \cdot -0.1388888888888889 + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \]

Alternative 9: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ 6.0 (* alpha 5.0))))
   (/ (/ (+ 1.0 alpha) (+ beta 3.0)) (+ beta 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d0) then
        tmp = (1.0d0 + alpha) / ((alpha + 2.0d0) * (6.0d0 + (alpha * 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + 3.0d0)) / (beta + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(6.0 + Float64(alpha * 5.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + 3.0)) / Float64(beta + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = (1.0 + alpha) / ((alpha + 2.0) * (6.0 + (alpha * 5.0)));
	else
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(6.0 + N[(alpha * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*r/ 99.2%

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

      11. associate-*r/ 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in alpha around 0 70.5%

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

   5. Taylor expanded in beta around 0 80.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(6 + 5 \cdot \alpha\right) \cdot \left(2 + \alpha\right)}} \]

   if 1.69999999999999996 < beta

   1. Initial program 83.4%

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

      1. associate-/l/ 80.3%

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

      2. associate-+l+ 80.3%

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

      3. +-commutative 80.3%

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

      4. associate-+r+ 80.3%

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

      5. associate-+l+ 80.3%

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

      6. distribute-rgt1-in 80.3%

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

      7. *-rgt-identity 80.3%

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

      8. distribute-lft-out 80.3%

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

      9. +-commutative 80.3%

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

      10. associate-*l/ 87.8%

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

      11. *-commutative 87.8%

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

      12. associate-*r/ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in alpha around 0 83.8%

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

   5. Taylor expanded in beta around inf 83.7%

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

      1. expm1-log1p-u 83.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-udef 46.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]

      3. un-div-inv 46.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 46.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 + \alpha}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1 \]

   7. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 83.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 83.7%

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

      3. associate-/r* 84.9%

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

   9. Simplified 84.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \]

Alternative 10: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ (/ (+ 1.0 alpha) (+ beta 3.0)) (+ beta 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / (beta + 3.0d0)) / (beta + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	else:
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + 3.0)) / Float64(beta + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	else
		tmp = ((1.0 + alpha) / (beta + 3.0)) / (beta + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.2], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 93.8%

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

   3. Simplified 93.8%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 69.8%

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

       1. associate-/r* 69.8%

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

   12. Simplified 69.8%

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

   13. Taylor expanded in beta around 0 68.4%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   if 1.19999999999999996 < beta

   1. Initial program 83.4%

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

      1. associate-/l/ 80.3%

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

      2. associate-+l+ 80.3%

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

      3. +-commutative 80.3%

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

      4. associate-+r+ 80.3%

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

      5. associate-+l+ 80.3%

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

      6. distribute-rgt1-in 80.3%

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

      7. *-rgt-identity 80.3%

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

      8. distribute-lft-out 80.3%

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

      9. +-commutative 80.3%

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

      10. associate-*l/ 87.8%

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

      11. *-commutative 87.8%

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

      12. associate-*r/ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in alpha around 0 83.8%

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

   5. Taylor expanded in beta around inf 83.7%

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

      1. expm1-log1p-u 83.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-udef 46.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\alpha + 1\right) \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]

      3. un-div-inv 46.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 46.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 + \alpha}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1 \]

   7. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 83.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 83.7%

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

      3. associate-/r* 84.9%

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

   9. Simplified 84.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 3}}{\beta + 2}\\ \end{array} \]

Alternative 11: 93.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.7)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (if (<= beta 2e+167)
     (* (/ 1.0 beta) (/ 1.0 beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 2e+167) {
		tmp = (1.0 / beta) * (1.0 / beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.7d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else if (beta <= 2d+167) then
        tmp = (1.0d0 / beta) * (1.0d0 / beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 2e+167) {
		tmp = (1.0 / beta) * (1.0 / beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.7:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	elif beta <= 2e+167:
		tmp = (1.0 / beta) * (1.0 / beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.7)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	elseif (beta <= 2e+167)
		tmp = Float64(Float64(1.0 / beta) * Float64(1.0 / beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.7)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	elseif (beta <= 2e+167)
		tmp = (1.0 / beta) * (1.0 / beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.7], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2e+167], N[(N[(1.0 / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.7000000000000002

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 93.8%

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

   3. Simplified 93.8%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 69.8%

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

       1. associate-/r* 69.8%

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

   12. Simplified 69.8%

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

   13. Taylor expanded in beta around 0 68.4%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   if 3.7000000000000002 < beta < 2.0000000000000001e167

   1. Initial program 97.2%

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

      1. associate-/l/ 91.6%

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

      2. associate-+l+ 91.6%

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

      3. +-commutative 91.6%

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

      4. associate-+r+ 91.6%

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

      5. associate-+l+ 91.6%

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

      6. distribute-rgt1-in 91.6%

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

      7. *-rgt-identity 91.6%

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

      8. distribute-lft-out 91.6%

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

      9. +-commutative 91.6%

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

      10. associate-*l/ 94.1%

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

      11. *-commutative 94.1%

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

      12. associate-*r/ 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in beta around inf 85.9%

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

      1. unpow2 85.9%

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

   6. Simplified 85.9%

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

   7. Taylor expanded in alpha around 0 79.3%

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

      1. unpow2 79.3%

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

   9. Simplified 79.3%

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

       1. inv-pow 79.3%

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

       2. unpow-prod-down 79.3%

          \[\leadsto \color{blue}{{\beta}^{-1} \cdot {\beta}^{-1}} \]

       3. inv-pow 79.3%

          \[\leadsto \color{blue}{\frac{1}{\beta}} \cdot {\beta}^{-1} \]

       4. inv-pow 79.3%

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

   11. Applied egg-rr 79.3%

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

   if 2.0000000000000001e167 < beta

   1. Initial program 69.6%

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

      1. associate-/l/ 69.0%

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

      2. associate-+l+ 69.0%

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

      3. +-commutative 69.0%

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

      4. associate-+r+ 69.0%

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

      5. associate-+l+ 69.0%

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

      6. distribute-rgt1-in 69.0%

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

      7. *-rgt-identity 69.0%

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

      8. distribute-lft-out 69.0%

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

      9. +-commutative 69.0%

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

      10. associate-*l/ 81.5%

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

      11. *-commutative 81.5%

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

      12. associate-*r/ 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in beta around inf 81.5%

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

      1. unpow2 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in alpha around inf 81.5%

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

      1. unpow2 81.5%

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

   9. Simplified 81.5%

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

   10. Taylor expanded in alpha around 0 81.5%

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

       1. unpow2 81.5%

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

       2. associate-/r* 84.9%

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

   12. Simplified 84.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12: 96.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (if (<= beta 1.35e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.8d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else if (beta <= 1.35d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	elif beta <= 1.35e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	elseif (beta <= 1.35e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 93.8%

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

   3. Simplified 93.8%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 69.8%

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

       1. associate-/r* 69.8%

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

   12. Simplified 69.8%

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

   13. Taylor expanded in beta around 0 68.4%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   if 3.7999999999999998 < beta < 1.35000000000000003e154

   1. Initial program 99.5%

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

      1. associate-/l/ 95.0%

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

      2. associate-+l+ 95.0%

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

      3. +-commutative 95.0%

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

      4. associate-+r+ 95.0%

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

      5. associate-+l+ 95.0%

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

      6. distribute-rgt1-in 95.0%

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

      7. *-rgt-identity 95.0%

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

      8. distribute-lft-out 95.1%

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

      9. +-commutative 95.1%

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

      10. associate-*l/ 95.1%

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

      11. *-commutative 95.1%

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

      12. associate-*r/ 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in beta around inf 86.3%

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

      1. unpow2 86.3%

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

   6. Simplified 86.3%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 69.4%

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

      1. associate-/l/ 67.6%

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

      2. associate-+l+ 67.6%

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

      3. +-commutative 67.6%

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

      4. associate-+r+ 67.6%

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

      5. associate-+l+ 67.6%

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

      6. distribute-rgt1-in 67.6%

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

      7. *-rgt-identity 67.6%

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

      8. distribute-lft-out 67.6%

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

      9. +-commutative 67.6%

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

      10. associate-*l/ 81.4%

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

      11. *-commutative 81.4%

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

      12. associate-*r/ 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in beta around inf 81.4%

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

      1. unpow2 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in alpha around inf 81.4%

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

      1. unpow2 81.4%

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

   9. Simplified 81.4%

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

   10. Taylor expanded in alpha around 0 81.4%

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

       1. unpow2 81.4%

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

       2. associate-/r* 82.5%

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

   12. Simplified 82.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13: 94.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.7)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (if (<= beta 1.4e+154) (/ 1.0 (* beta beta)) (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 1.4e+154) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.7d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else if (beta <= 1.4d+154) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.7) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 1.4e+154) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.7:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	elif beta <= 1.4e+154:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.7)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	elseif (beta <= 1.4e+154)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.7)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	elseif (beta <= 1.4e+154)
		tmp = 1.0 / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.7], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.4e+154], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.7000000000000002

   1. Initial program 99.9%

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

      1. associate-/l/ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. associate-+l+ 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-*l/ 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-*r/ 93.8%

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

   3. Simplified 93.8%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-/r* 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in alpha around 0 69.8%

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

       1. associate-/r* 69.8%

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

   12. Simplified 69.8%

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

   13. Taylor expanded in beta around 0 68.4%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   if 3.7000000000000002 < beta < 1.4e154

   1. Initial program 99.5%

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

      1. associate-/l/ 95.0%

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

      2. associate-+l+ 95.0%

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

      3. +-commutative 95.0%

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

      4. associate-+r+ 95.0%

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

      5. associate-+l+ 95.0%

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

      6. distribute-rgt1-in 95.0%

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

      7. *-rgt-identity 95.0%

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

      8. distribute-lft-out 95.1%

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

      9. +-commutative 95.1%

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

      10. associate-*l/ 95.1%

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

      11. *-commutative 95.1%

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

      12. associate-*r/ 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in beta around inf 86.3%

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

      1. unpow2 86.3%

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

   6. Simplified 86.3%

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

   7. Taylor expanded in alpha around 0 79.2%

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

      1. unpow2 79.2%

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

   9. Simplified 79.2%

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

   if 1.4e154 < beta

   1. Initial program 69.4%

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

      1. associate-/l/ 67.6%

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

      2. associate-+l+ 67.6%

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

      3. +-commutative 67.6%

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

      4. associate-+r+ 67.6%

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

      5. associate-+l+ 67.6%

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

      6. distribute-rgt1-in 67.6%

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

      7. *-rgt-identity 67.6%

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

      8. distribute-lft-out 67.6%

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

      9. +-commutative 67.6%

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

      10. associate-*l/ 81.4%

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

      11. *-commutative 81.4%

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

      12. associate-*r/ 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in beta around inf 81.4%

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

      1. unpow2 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in alpha around inf 81.4%

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

      1. unpow2 81.4%

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

   9. Simplified 81.4%

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

   10. Taylor expanded in alpha around 0 81.4%

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

       1. unpow2 81.4%

          \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]

       2. associate-/r* 82.5%

          \[\leadsto \color{blue}{\frac{\frac{\alpha}{\beta}}{\beta}} \]

   12. Simplified 82.5%

       \[\leadsto \color{blue}{\frac{\frac{\alpha}{\beta}}{\beta}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14: 91.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.2%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 99.2%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 99.2%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 99.2%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 99.2%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 99.2%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 99.2%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 99.2%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 99.2%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 99.2%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 99.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 93.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.2%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. +-commutative 99.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 99.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. *-commutative 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \alpha\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 99.2%

         \[\leadsto \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(\alpha + 1\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-*r/ 99.2%

         \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      5. associate-/r* 99.8%

         \[\leadsto \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\color{blue}{\beta + 1}}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      7. associate-+r+ 99.8%

         \[\leadsto \frac{\beta + 1}{\color{blue}{\left(\alpha + \beta\right) + 2}} \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\beta + 1}{\color{blue}{\left(\beta + \alpha\right)} + 2} \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      9. +-commutative 99.8%

         \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\alpha + \beta\right) + 2}}}{\alpha + \left(\beta + 3\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\alpha + \left(\beta + 3\right)} \]

      12. +-commutative 99.8%

          \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\left(\beta + 3\right) + \alpha}} \]
   8. Step-by-step derivation

      1. div-inv 99.8%

         \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\left(\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{1}{\left(\beta + 3\right) + \alpha}\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{1 + \alpha}{\color{blue}{\left(\alpha + \beta\right)} + 2} \cdot \frac{1}{\left(\beta + 3\right) + \alpha}\right) \]

      3. associate-+r+ 99.8%

         \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{\left(\beta + 3\right) + \alpha}\right) \]

      4. +-commutative 99.8%

         \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\right)} \]

   10. Taylor expanded in alpha around 0 69.8%

       \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}} \]
   11. Step-by-step derivation

       1. associate-/r* 69.8%

          \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\frac{\frac{1}{\beta + 3}}{\beta + 2}} \]

   12. Simplified 69.8%

       \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\frac{\frac{1}{\beta + 3}}{\beta + 2}} \]

   13. Taylor expanded in beta around 0 68.4%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   if 4.20000000000000018 < beta

   1. Initial program 83.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-+l+ 80.3%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative 80.3%

         \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+r+ 80.3%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+l+ 80.3%

         \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. distribute-rgt1-in 80.3%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity 80.3%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-lft-out 80.3%

         \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. +-commutative 80.3%

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ 87.8%

          \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. *-commutative 87.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      12. associate-*r/ 87.8%

          \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 83.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 83.7%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around 0 80.4%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 80.4%

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15: 45.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\alpha + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ alpha 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

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 = 0.16666666666666666d0 / (alpha + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (alpha + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (alpha + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(alpha + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (alpha + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 93.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 93.0%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 93.0%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 93.0%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ 95.4%

       \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. *-commutative 95.4%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. associate-*r/ 91.8%

       \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 91.8%

   \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 95.4%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   2. +-commutative 95.4%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

5. Applied egg-rr 95.4%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative 95.4%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   2. *-commutative 95.4%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \alpha\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   3. +-commutative 95.4%

      \[\leadsto \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(\alpha + 1\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   4. associate-*r/ 95.4%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   5. associate-/r* 99.8%

      \[\leadsto \frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]

   6. +-commutative 99.8%

      \[\leadsto \frac{\color{blue}{\beta + 1}}{\alpha + \left(\beta + 2\right)} \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

   7. associate-+r+ 99.8%

      \[\leadsto \frac{\beta + 1}{\color{blue}{\left(\alpha + \beta\right) + 2}} \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

   8. +-commutative 99.8%

      \[\leadsto \frac{\beta + 1}{\color{blue}{\left(\beta + \alpha\right)} + 2} \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

   9. +-commutative 99.8%

      \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]

   10. associate-+r+ 99.8%

       \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\alpha + \beta\right) + 2}}}{\alpha + \left(\beta + 3\right)} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\alpha + \left(\beta + 3\right)} \]

   12. +-commutative 99.8%

       \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2}}{\left(\beta + 3\right) + \alpha}} \]
8. Step-by-step derivation

   1. div-inv 99.7%

      \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\left(\frac{1 + \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{1}{\left(\beta + 3\right) + \alpha}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{1 + \alpha}{\color{blue}{\left(\alpha + \beta\right)} + 2} \cdot \frac{1}{\left(\beta + 3\right) + \alpha}\right) \]

   3. associate-+r+ 99.7%

      \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{\left(\beta + 3\right) + \alpha}\right) \]

   4. +-commutative 99.7%

      \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\color{blue}{\alpha + \left(\beta + 3\right)}}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\right)} \]

10. Taylor expanded in alpha around 0 73.3%

    \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}} \]
11. Step-by-step derivation

    1. associate-/r* 73.5%

       \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\frac{\frac{1}{\beta + 3}}{\beta + 2}} \]

12. Simplified 73.5%

    \[\leadsto \frac{\beta + 1}{\left(\beta + \alpha\right) + 2} \cdot \color{blue}{\frac{\frac{1}{\beta + 3}}{\beta + 2}} \]

13. Taylor expanded in beta around 0 47.6%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

14. Final simplification 47.6%

    \[\leadsto \frac{0.16666666666666666}{\alpha + 2} \]

Alternative 16: 10.7% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.16666666666666666 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.16666666666666666)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666;
}

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 = 0.16666666666666666d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.16666666666666666
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.16666666666666666

Derivation

1. Initial program 94.5%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 93.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-+l+ 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative 93.0%

      \[\leadsto \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+r+ 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+l+ 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. distribute-rgt1-in 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity 93.0%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-lft-out 93.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. +-commutative 93.0%

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ 95.4%

       \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. *-commutative 95.4%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   12. associate-*r/ 91.8%

       \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 91.8%

   \[\leadsto \color{blue}{\left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

4. Taylor expanded in alpha around 0 73.8%

   \[\leadsto \left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}} \]

5. Taylor expanded in alpha around inf 24.9%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 11.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\beta + 3}}{\beta + 2}} \]

   2. +-commutative 11.8%

      \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\beta + 3}}{\beta + 2} \]

7. Simplified 11.8%

   \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\beta + 3}}{\beta + 2}} \]

8. Taylor expanded in beta around 0 10.9%

   \[\leadsto \color{blue}{0.16666666666666666} \]

9. Final simplification 10.9%

   \[\leadsto 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023257 
(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)))