Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 27.2s

Alternatives: 28

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

   1. associate-/l/ 90.7%

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

   2. associate-/r* 83.2%

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

   3. +-commutative 83.2%

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

   4. associate-+r+ 83.2%

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

   5. +-commutative 83.2%

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

   6. associate-+r+ 83.2%

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

   7. associate-+r+ 83.2%

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

   8. distribute-rgt1-in 83.2%

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

   9. +-commutative 83.2%

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

   10. *-commutative 83.2%

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

   11. distribute-rgt1-in 83.2%

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

   12. +-commutative 83.2%

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

   13. times-frac 95.9%

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

3. Simplified 95.9%

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

   1. expm1-log1p-u 95.9%

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

   2. expm1-udef 68.9%

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

   3. *-commutative 68.9%

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

   4. +-commutative 68.9%

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

5. Applied egg-rr 68.9%

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

   1. expm1-def 95.9%

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

   2. expm1-log1p 95.9%

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

   3. associate-*r/ 95.8%

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

   4. *-commutative 95.8%

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

   5. associate-*r/ 90.7%

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

   6. times-frac 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t_0} \cdot \left(\frac{1}{\beta} - \frac{2 \cdot \left(\alpha + 2\right)}{\beta \cdot \beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 9e+33)
     (/ (/ (/ (+ 1.0 beta) (+ beta 3.0)) (+ 2.0 beta)) t_0)
     (*
      (/ (+ 1.0 alpha) t_0)
      (- (/ 1.0 beta) (/ (* 2.0 (+ alpha 2.0)) (* beta beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 9e+33) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 / beta) - ((2.0 * (alpha + 2.0)) / (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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 9d+33) then
        tmp = (((1.0d0 + beta) / (beta + 3.0d0)) / (2.0d0 + beta)) / t_0
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 / beta) - ((2.0d0 * (alpha + 2.0d0)) / (beta * beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 9e+33) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 / beta) - ((2.0 * (alpha + 2.0)) / (beta * beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 9e+33:
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / t_0
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 / beta) - ((2.0 * (alpha + 2.0)) / (beta * beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 9e+33)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(beta + 3.0)) / Float64(2.0 + beta)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 / beta) - Float64(Float64(2.0 * Float64(alpha + 2.0)) / Float64(beta * beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 9e+33)
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / t_0;
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 / beta) - ((2.0 * (alpha + 2.0)) / (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_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9e+33], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 / beta), $MachinePrecision] - N[(N[(2.0 * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.0000000000000001e33

   1. Initial program 99.3%

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

      1. associate-/l/ 99.0%

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

      2. associate-/r* 95.4%

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

      3. +-commutative 95.4%

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

      4. associate-+r+ 95.4%

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

      5. +-commutative 95.4%

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

      6. associate-+r+ 95.4%

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

      7. associate-+r+ 95.4%

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

      8. distribute-rgt1-in 95.4%

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

      9. +-commutative 95.4%

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

      10. *-commutative 95.4%

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

      11. distribute-rgt1-in 95.4%

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

      12. +-commutative 95.4%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 77.3%

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

      3. *-commutative 77.3%

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

      4. +-commutative 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.0%

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

      6. times-frac 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 68.4%

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

      1. associate-/r* 68.4%

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

   10. Simplified 68.4%

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

   if 9.0000000000000001e33 < beta

   1. Initial program 76.6%

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

      1. associate-/l/ 72.9%

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

      2. associate-/r* 57.0%

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

      3. +-commutative 57.0%

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

      4. associate-+r+ 57.0%

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

      5. +-commutative 57.0%

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

      6. associate-+r+ 57.0%

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

      7. associate-+r+ 57.0%

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

      8. distribute-rgt1-in 57.0%

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

      9. +-commutative 57.0%

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

      10. *-commutative 57.0%

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

      11. distribute-rgt1-in 57.0%

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

      12. +-commutative 57.0%

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

      13. times-frac 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in beta around inf 83.1%

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

      1. +-commutative 83.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(\frac{1}{\beta} + -1 \cdot \frac{4 + 2 \cdot \alpha}{{\beta}^{2}}\right)} \]

      2. mul-1-neg 83.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(\frac{1}{\beta} + \color{blue}{\left(-\frac{4 + 2 \cdot \alpha}{{\beta}^{2}}\right)}\right) \]

      3. unsub-neg 83.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(\frac{1}{\beta} - \frac{4 + 2 \cdot \alpha}{{\beta}^{2}}\right)} \]

      4. metadata-eval 83.1%

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

      5. distribute-lft-in 83.1%

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

      6. unpow2 83.1%

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

   6. Simplified 83.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(\frac{1}{\beta} - \frac{2 \cdot \left(\alpha + 2\right)}{\beta \cdot \beta}\right)\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{t_0}{1 + \beta} \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 5e+17)
     (/ 1.0 (* (/ t_0 (+ 1.0 beta)) (* (+ 2.0 beta) (+ beta 3.0))))
     (/ (* (/ (+ 1.0 alpha) t_0) (+ 1.0 (/ (- -2.0 alpha) beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 5e+17) {
		tmp = 1.0 / ((t_0 / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 5d+17) then
        tmp = 1.0d0 / ((t_0 / (1.0d0 + beta)) * ((2.0d0 + beta) * (beta + 3.0d0)))
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 + (((-2.0d0) - alpha) / beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 5e+17) {
		tmp = 1.0 / ((t_0 / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 5e+17:
		tmp = 1.0 / ((t_0 / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)))
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 5e+17)
		tmp = Float64(1.0 / Float64(Float64(t_0 / Float64(1.0 + beta)) * Float64(Float64(2.0 + beta) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 + Float64(Float64(-2.0 - alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 5e+17)
		tmp = 1.0 / ((t_0 / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+17], N[(1.0 / N[(N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e17

   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/ 99.5%

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

      2. associate-/r* 96.3%

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

      3. +-commutative 96.3%

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

      4. associate-+r+ 96.3%

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

      5. +-commutative 96.3%

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

      6. associate-+r+ 96.3%

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

      7. associate-+r+ 96.3%

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

      8. distribute-rgt1-in 96.3%

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

      9. +-commutative 96.3%

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

      10. *-commutative 96.3%

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

      11. distribute-rgt1-in 96.3%

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

      12. +-commutative 96.3%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.1%

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

      3. *-commutative 81.1%

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

      4. +-commutative 81.1%

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

   5. Applied egg-rr 81.1%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 67.0%

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

      1. clear-num 67.0%

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

      2. inv-pow 67.0%

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

      3. +-commutative 67.0%

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

      4. +-commutative 67.0%

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

   10. Applied egg-rr 67.0%

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

       1. unpow-1 67.0%

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

       2. associate-/r/ 67.1%

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

       3. +-commutative 67.1%

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

       4. +-commutative 67.1%

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

       5. +-commutative 67.1%

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

       6. *-commutative 67.1%

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

   12. Simplified 67.1%

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

   if 5e17 < beta

   1. Initial program 78.1%

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

      1. associate-/l/ 74.7%

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

      2. associate-/r* 59.5%

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

      3. +-commutative 59.5%

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

      4. associate-+r+ 59.5%

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

      5. +-commutative 59.5%

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

      6. associate-+r+ 59.5%

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

      7. associate-+r+ 59.5%

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

      8. distribute-rgt1-in 59.5%

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

      9. +-commutative 59.5%

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

      10. *-commutative 59.5%

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

      11. distribute-rgt1-in 59.5%

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

      12. +-commutative 59.5%

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

      13. times-frac 89.3%

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

   3. Simplified 89.3%

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

      1. expm1-log1p-u 89.3%

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

      2. expm1-udef 46.8%

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

      3. *-commutative 46.8%

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

      4. +-commutative 46.8%

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

   5. Applied egg-rr 46.8%

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

      1. expm1-def 89.3%

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

      2. expm1-log1p 89.3%

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

      3. associate-*r/ 89.2%

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

      4. *-commutative 89.2%

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

      5. associate-*r/ 74.7%

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

      6. times-frac 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around inf 82.9%

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

      1. associate-*r/ 82.9%

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

      2. distribute-lft-in 82.9%

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

      3. metadata-eval 82.9%

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

      4. mul-1-neg 82.9%

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

      5. unsub-neg 82.9%

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

   10. Simplified 82.9%

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

4. Final simplification 72.7%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 210000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{t_0}}{t_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 210000000.0)
     (/ (/ (+ 1.0 (+ alpha beta)) t_0) (* t_0 (+ (+ alpha beta) 3.0)))
     (/ (* (/ (+ 1.0 alpha) t_0) (+ 1.0 (/ (- -2.0 alpha) beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 210000000.0) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * ((alpha + beta) + 3.0));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 210000000.0d0) then
        tmp = ((1.0d0 + (alpha + beta)) / t_0) / (t_0 * ((alpha + beta) + 3.0d0))
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 + (((-2.0d0) - alpha) / beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 210000000.0) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * ((alpha + beta) + 3.0));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 210000000.0:
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * ((alpha + beta) + 3.0))
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 210000000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + beta)) / t_0) / Float64(t_0 * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 + Float64(Float64(-2.0 - alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 210000000.0)
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * ((alpha + beta) + 3.0));
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 210000000.0], N[(N[(N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.1e8

   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/ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-+l+ 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in beta around 0 99.5%

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

   if 2.1e8 < beta

   1. Initial program 78.3%

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

      1. associate-/l/ 75.0%

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

      2. associate-/r* 60.0%

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

      3. +-commutative 60.0%

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

      4. associate-+r+ 60.0%

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

      5. +-commutative 60.0%

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

      6. associate-+r+ 60.0%

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

      7. associate-+r+ 60.0%

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

      8. distribute-rgt1-in 60.0%

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

      9. +-commutative 60.0%

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

      10. *-commutative 60.0%

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

      11. distribute-rgt1-in 60.0%

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

      12. +-commutative 60.0%

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

      13. times-frac 89.4%

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

   3. Simplified 89.4%

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

      1. expm1-log1p-u 89.4%

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

      2. expm1-udef 46.4%

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

      3. *-commutative 46.4%

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

      4. +-commutative 46.4%

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

   5. Applied egg-rr 46.4%

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

      1. expm1-def 89.4%

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

      2. expm1-log1p 89.4%

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

      3. associate-*r/ 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-*r/ 75.0%

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

      6. times-frac 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around inf 83.1%

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

      1. associate-*r/ 83.1%

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

      2. distribute-lft-in 83.1%

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

      3. metadata-eval 83.1%

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

      4. mul-1-neg 83.1%

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

      5. unsub-neg 83.1%

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

   10. Simplified 83.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 210000000:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

   1. associate-/l/ 90.7%

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

   2. associate-/r* 83.2%

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

   3. +-commutative 83.2%

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

   4. associate-+r+ 83.2%

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

   5. +-commutative 83.2%

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

   6. associate-+r+ 83.2%

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

   7. associate-+r+ 83.2%

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

   8. distribute-rgt1-in 83.2%

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

   9. +-commutative 83.2%

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

   10. *-commutative 83.2%

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

   11. distribute-rgt1-in 83.2%

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

   12. +-commutative 83.2%

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

   13. times-frac 95.9%

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

3. Simplified 95.9%

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

   1. expm1-log1p-u 95.9%

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

   2. expm1-udef 68.9%

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

   3. *-commutative 68.9%

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

   4. +-commutative 68.9%

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

5. Applied egg-rr 68.9%

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

   1. expm1-def 95.9%

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

   2. expm1-log1p 95.9%

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

   3. associate-*r/ 95.8%

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

   4. *-commutative 95.8%

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

   5. associate-*r/ 90.7%

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

   6. times-frac 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. associate-/l* 99.6%

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

   3. associate-+l+ 99.6%

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

9. Applied egg-rr 99.6%

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

    1. *-lft-identity 99.6%

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

    2. associate-/r/ 99.8%

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

    3. associate-+r+ 99.8%

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

    4. +-commutative 99.8%

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

    5. associate-+r+ 99.8%

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

    6. +-commutative 99.8%

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

    7. +-commutative 99.8%

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

    8. +-commutative 99.8%

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

11. Simplified 99.8%

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

    1. clear-num 99.8%

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

    2. inv-pow 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-+r+ 99.8%

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

13. Applied egg-rr 99.8%

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

    1. unpow-1 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

15. Simplified 99.8%

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

16. Taylor expanded in alpha around 0 72.9%

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

17. Final simplification 72.9%

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

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

   1. associate-/l/ 90.7%

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

   2. associate-/r* 83.2%

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

   3. +-commutative 83.2%

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

   4. associate-+r+ 83.2%

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

   5. +-commutative 83.2%

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

   6. associate-+r+ 83.2%

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

   7. associate-+r+ 83.2%

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

   8. distribute-rgt1-in 83.2%

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

   9. +-commutative 83.2%

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

   10. *-commutative 83.2%

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

   11. distribute-rgt1-in 83.2%

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

   12. +-commutative 83.2%

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

   13. times-frac 95.9%

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

3. Simplified 95.9%

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

   1. expm1-log1p-u 95.9%

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

   2. expm1-udef 68.9%

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

   3. *-commutative 68.9%

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

   4. +-commutative 68.9%

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

5. Applied egg-rr 68.9%

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

   1. expm1-def 95.9%

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

   2. expm1-log1p 95.9%

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

   3. *-commutative 95.9%

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

   4. associate-*r/ 95.9%

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

   5. times-frac 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 7: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2e+17)
   (/
    1.0
    (* (/ (+ alpha (+ 2.0 beta)) (+ 1.0 beta)) (* (+ 2.0 beta) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+17) {
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d+17) then
        tmp = 1.0d0 / (((alpha + (2.0d0 + beta)) / (1.0d0 + beta)) * ((2.0d0 + beta) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+17) {
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2e+17:
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2e+17)
		tmp = Float64(1.0 / Float64(Float64(Float64(alpha + Float64(2.0 + beta)) / Float64(1.0 + beta)) * Float64(Float64(2.0 + beta) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2e+17)
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2e+17], N[(1.0 / N[(N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2e17

   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/ 99.5%

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

      2. associate-/r* 96.3%

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

      3. +-commutative 96.3%

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

      4. associate-+r+ 96.3%

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

      5. +-commutative 96.3%

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

      6. associate-+r+ 96.3%

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

      7. associate-+r+ 96.3%

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

      8. distribute-rgt1-in 96.3%

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

      9. +-commutative 96.3%

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

      10. *-commutative 96.3%

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

      11. distribute-rgt1-in 96.3%

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

      12. +-commutative 96.3%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.1%

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

      3. *-commutative 81.1%

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

      4. +-commutative 81.1%

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

   5. Applied egg-rr 81.1%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 67.0%

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

      1. clear-num 67.0%

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

      2. inv-pow 67.0%

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

      3. +-commutative 67.0%

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

      4. +-commutative 67.0%

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

   10. Applied egg-rr 67.0%

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

       1. unpow-1 67.0%

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

       2. associate-/r/ 67.1%

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

       3. +-commutative 67.1%

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

       4. +-commutative 67.1%

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

       5. +-commutative 67.1%

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

       6. *-commutative 67.1%

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

   12. Simplified 67.1%

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

   if 5.2e17 < beta

   1. Initial program 78.1%

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

      1. associate-/l/ 74.7%

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

      2. associate-/r* 59.5%

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

      3. +-commutative 59.5%

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

      4. associate-+r+ 59.5%

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

      5. +-commutative 59.5%

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

      6. associate-+r+ 59.5%

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

      7. associate-+r+ 59.5%

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

      8. distribute-rgt1-in 59.5%

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

      9. +-commutative 59.5%

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

      10. *-commutative 59.5%

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

      11. distribute-rgt1-in 59.5%

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

      12. +-commutative 59.5%

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

      13. times-frac 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in beta around inf 83.0%

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

      1. associate-*l/ 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

      4. associate-+l+ 83.2%

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

      5. +-commutative 83.2%

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

      6. associate-+l+ 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-*r/ 83.1%

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

      2. *-rgt-identity 83.1%

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

   8. Simplified 83.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 8: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \left(\frac{1}{\beta} - \frac{2}{\beta} \cdot \frac{\alpha + 2}{\beta}\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8e+18)
   (/
    1.0
    (* (/ (+ alpha (+ 2.0 beta)) (+ 1.0 beta)) (* (+ 2.0 beta) (+ beta 3.0))))
   (/
    (* (+ 1.0 alpha) (- (/ 1.0 beta) (* (/ 2.0 beta) (/ (+ alpha 2.0) beta))))
    beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8e+18) {
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 / beta) - ((2.0 / beta) * ((alpha + 2.0) / 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 <= 7.8d+18) then
        tmp = 1.0d0 / (((alpha + (2.0d0 + beta)) / (1.0d0 + beta)) * ((2.0d0 + beta) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 / beta) - ((2.0d0 / beta) * ((alpha + 2.0d0) / beta)))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8e+18) {
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 / beta) - ((2.0 / beta) * ((alpha + 2.0) / beta)))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.8e+18:
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) * ((1.0 / beta) - ((2.0 / beta) * ((alpha + 2.0) / beta)))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8e+18)
		tmp = Float64(1.0 / Float64(Float64(Float64(alpha + Float64(2.0 + beta)) / Float64(1.0 + beta)) * Float64(Float64(2.0 + beta) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 / beta) - Float64(Float64(2.0 / beta) * Float64(Float64(alpha + 2.0) / beta)))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.8e+18)
		tmp = 1.0 / (((alpha + (2.0 + beta)) / (1.0 + beta)) * ((2.0 + beta) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) * ((1.0 / beta) - ((2.0 / beta) * ((alpha + 2.0) / 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, 7.8e+18], N[(1.0 / N[(N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 / beta), $MachinePrecision] - N[(N[(2.0 / beta), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8e18

   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/ 99.5%

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

      2. associate-/r* 96.3%

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

      3. +-commutative 96.3%

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

      4. associate-+r+ 96.3%

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

      5. +-commutative 96.3%

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

      6. associate-+r+ 96.3%

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

      7. associate-+r+ 96.3%

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

      8. distribute-rgt1-in 96.3%

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

      9. +-commutative 96.3%

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

      10. *-commutative 96.3%

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

      11. distribute-rgt1-in 96.3%

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

      12. +-commutative 96.3%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.1%

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

      3. *-commutative 81.1%

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

      4. +-commutative 81.1%

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

   5. Applied egg-rr 81.1%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 67.0%

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

      1. clear-num 67.0%

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

      2. inv-pow 67.0%

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

      3. +-commutative 67.0%

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

      4. +-commutative 67.0%

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

   10. Applied egg-rr 67.0%

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

       1. unpow-1 67.0%

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

       2. associate-/r/ 67.1%

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

       3. +-commutative 67.1%

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

       4. +-commutative 67.1%

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

       5. +-commutative 67.1%

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

       6. *-commutative 67.1%

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

   12. Simplified 67.1%

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

   if 7.8e18 < beta

   1. Initial program 78.1%

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

      1. associate-/l/ 74.7%

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

      2. associate-/r* 59.5%

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

      3. +-commutative 59.5%

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

      4. associate-+r+ 59.5%

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

      5. +-commutative 59.5%

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

      6. associate-+r+ 59.5%

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

      7. associate-+r+ 59.5%

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

      8. distribute-rgt1-in 59.5%

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

      9. +-commutative 59.5%

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

      10. *-commutative 59.5%

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

      11. distribute-rgt1-in 59.5%

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

      12. +-commutative 59.5%

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

      13. times-frac 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in beta around inf 85.3%

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

      1. associate-*l/ 85.3%

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

      2. +-commutative 85.3%

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

      3. *-commutative 85.3%

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

      4. associate-+r+ 85.3%

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

      5. +-commutative 85.3%

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

      6. associate-+l+ 85.3%

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

      7. +-commutative 85.3%

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

   6. Applied egg-rr 85.3%

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

   7. Taylor expanded in beta around inf 82.7%

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

      1. +-commutative 82.7%

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \color{blue}{\left(\frac{1}{\beta} + -1 \cdot \frac{4 + 2 \cdot \alpha}{{\beta}^{2}}\right)}}{\beta} \]

      2. mul-1-neg 82.7%

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \left(\frac{1}{\beta} + \color{blue}{\left(-\frac{4 + 2 \cdot \alpha}{{\beta}^{2}}\right)}\right)}{\beta} \]

      3. unsub-neg 82.7%

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \color{blue}{\left(\frac{1}{\beta} - \frac{4 + 2 \cdot \alpha}{{\beta}^{2}}\right)}}{\beta} \]

      4. metadata-eval 82.7%

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

      5. distribute-lft-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.6%

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

      8. +-commutative 82.6%

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

   9. Simplified 82.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(2 + \beta\right)}{1 + \beta} \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \left(\frac{1}{\beta} - \frac{2}{\beta} \cdot \frac{\alpha + 2}{\beta}\right)}{\beta}\\ \end{array} \]

Alternative 9: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 24:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 24.0)
     (/ (+ 1.0 alpha) (* t_0 (* (+ alpha 2.0) (+ alpha 3.0))))
     (/ (/ (- alpha -1.0) beta) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 24.0) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (1.0 + t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 24.0d0) then
        tmp = (1.0d0 + alpha) / (t_0 * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 24.0) {
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 24.0:
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((alpha - -1.0) / beta) / (1.0 + t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 24.0)
		tmp = Float64(Float64(1.0 + alpha) / Float64(t_0 * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 24.0)
		tmp = (1.0 + alpha) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((alpha - -1.0) / beta) / (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 24.0], N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 24

   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/ 99.5%

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

      2. associate-/r* 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-+r+ 96.2%

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

      5. +-commutative 96.2%

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

      6. associate-+r+ 96.2%

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

      7. associate-+r+ 96.2%

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

      8. distribute-rgt1-in 96.2%

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

      9. +-commutative 96.2%

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

      10. *-commutative 96.2%

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

      11. distribute-rgt1-in 96.2%

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

      12. +-commutative 96.2%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.5%

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

      3. *-commutative 81.5%

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

      4. +-commutative 81.5%

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

   5. Applied egg-rr 81.5%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in beta around 0 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/l/ 94.4%

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

      3. associate-+r+ 94.4%

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

      4. +-commutative 94.4%

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

      5. associate-+l+ 94.4%

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

      6. +-commutative 94.4%

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

   10. Applied egg-rr 94.4%

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

       1. *-lft-identity 94.4%

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

       2. +-commutative 94.4%

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

       3. *-commutative 94.4%

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

       4. +-commutative 94.4%

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

       5. +-commutative 94.4%

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

   12. Simplified 94.4%

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

   if 24 < beta

   1. Initial program 78.8%

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

   2. Taylor expanded in beta around -inf 81.8%

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

      1. associate-*r/ 81.8%

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

      2. mul-1-neg 81.8%

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

      3. sub-neg 81.8%

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

      4. mul-1-neg 81.8%

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

      5. distribute-neg-in 81.8%

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

      6. +-commutative 81.8%

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

      7. mul-1-neg 81.8%

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

      8. distribute-lft-in 81.8%

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

      9. metadata-eval 81.8%

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

      10. mul-1-neg 81.8%

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

      11. unsub-neg 81.8%

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

   4. Simplified 81.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 24:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 10: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.35e+18)
   (/ (/ (+ 1.0 beta) (+ alpha (+ 2.0 beta))) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35e+18) {
		tmp = ((1.0 + beta) / (alpha + (2.0 + beta))) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.35d+18) then
        tmp = ((1.0d0 + beta) / (alpha + (2.0d0 + beta))) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35e+18) {
		tmp = ((1.0 + beta) / (alpha + (2.0 + beta))) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.35e+18:
		tmp = ((1.0 + beta) / (alpha + (2.0 + beta))) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.35e+18)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(alpha + Float64(2.0 + beta))) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.35e+18)
		tmp = ((1.0 + beta) / (alpha + (2.0 + beta))) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.35e+18], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.35e18

   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/ 99.5%

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

      2. associate-/r* 96.3%

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

      3. +-commutative 96.3%

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

      4. associate-+r+ 96.3%

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

      5. +-commutative 96.3%

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

      6. associate-+r+ 96.3%

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

      7. associate-+r+ 96.3%

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

      8. distribute-rgt1-in 96.3%

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

      9. +-commutative 96.3%

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

      10. *-commutative 96.3%

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

      11. distribute-rgt1-in 96.3%

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

      12. +-commutative 96.3%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.1%

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

      3. *-commutative 81.1%

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

      4. +-commutative 81.1%

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

   5. Applied egg-rr 81.1%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 67.0%

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

      1. expm1-log1p-u 67.0%

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

      2. expm1-udef 78.7%

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

      3. associate-/l/ 78.7%

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

      4. +-commutative 78.7%

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

      5. +-commutative 78.7%

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

   10. Applied egg-rr 78.7%

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

       1. expm1-def 67.1%

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

       2. expm1-log1p 67.1%

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

       3. associate-/r* 67.0%

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

       4. +-commutative 67.0%

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

       5. +-commutative 67.0%

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

       6. +-commutative 67.0%

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

       7. *-commutative 67.0%

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

   12. Simplified 67.0%

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

   if 1.35e18 < beta

   1. Initial program 78.1%

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

      1. associate-/l/ 74.7%

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

      2. associate-/r* 59.5%

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

      3. +-commutative 59.5%

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

      4. associate-+r+ 59.5%

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

      5. +-commutative 59.5%

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

      6. associate-+r+ 59.5%

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

      7. associate-+r+ 59.5%

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

      8. distribute-rgt1-in 59.5%

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

      9. +-commutative 59.5%

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

      10. *-commutative 59.5%

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

      11. distribute-rgt1-in 59.5%

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

      12. +-commutative 59.5%

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

      13. times-frac 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in beta around inf 83.0%

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

      1. associate-*l/ 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

      4. associate-+l+ 83.2%

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

      5. +-commutative 83.2%

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

      6. associate-+l+ 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-*r/ 83.1%

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

      2. *-rgt-identity 83.1%

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

   8. Simplified 83.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 11: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4e+16)
   (/ (/ (/ (+ 1.0 beta) (+ beta 3.0)) (+ 2.0 beta)) (+ alpha (+ 2.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4e+16) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.4d+16) then
        tmp = (((1.0d0 + beta) / (beta + 3.0d0)) / (2.0d0 + beta)) / (alpha + (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4e+16) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.4e+16:
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (alpha + (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4e+16)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(beta + 3.0)) / Float64(2.0 + beta)) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.4e+16)
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (alpha + (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4e+16], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4e16

   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/ 99.5%

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

      2. associate-/r* 96.3%

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

      3. +-commutative 96.3%

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

      4. associate-+r+ 96.3%

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

      5. +-commutative 96.3%

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

      6. associate-+r+ 96.3%

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

      7. associate-+r+ 96.3%

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

      8. distribute-rgt1-in 96.3%

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

      9. +-commutative 96.3%

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

      10. *-commutative 96.3%

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

      11. distribute-rgt1-in 96.3%

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

      12. +-commutative 96.3%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.1%

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

      3. *-commutative 81.1%

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

      4. +-commutative 81.1%

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

   5. Applied egg-rr 81.1%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 67.0%

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

      1. associate-/r* 67.1%

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

   10. Simplified 67.1%

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

   if 3.4e16 < beta

   1. Initial program 78.1%

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

      1. associate-/l/ 74.7%

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

      2. associate-/r* 59.5%

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

      3. +-commutative 59.5%

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

      4. associate-+r+ 59.5%

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

      5. +-commutative 59.5%

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

      6. associate-+r+ 59.5%

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

      7. associate-+r+ 59.5%

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

      8. distribute-rgt1-in 59.5%

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

      9. +-commutative 59.5%

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

      10. *-commutative 59.5%

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

      11. distribute-rgt1-in 59.5%

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

      12. +-commutative 59.5%

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

      13. times-frac 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in beta around inf 83.0%

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

      1. associate-*l/ 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

      4. associate-+l+ 83.2%

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

      5. +-commutative 83.2%

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

      6. associate-+l+ 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-*r/ 83.1%

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

      2. *-rgt-identity 83.1%

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

   8. Simplified 83.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 12: 97.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{\frac{1 + \beta}{6 + \beta \cdot 5}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.8)
   (/ (/ (+ 1.0 beta) (+ 6.0 (* beta 5.0))) (+ alpha (+ 2.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.8d0) then
        tmp = ((1.0d0 + beta) / (6.0d0 + (beta * 5.0d0))) / (alpha + (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.8:
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.8)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(6.0 + Float64(beta * 5.0))) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.8)
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.8], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999982

   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/ 99.5%

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

      2. associate-/r* 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-+r+ 96.2%

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

      5. +-commutative 96.2%

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

      6. associate-+r+ 96.2%

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

      7. associate-+r+ 96.2%

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

      8. distribute-rgt1-in 96.2%

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

      9. +-commutative 96.2%

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

      10. *-commutative 96.2%

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

      11. distribute-rgt1-in 96.2%

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

      12. +-commutative 96.2%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.5%

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

      3. *-commutative 81.5%

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

      4. +-commutative 81.5%

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

   5. Applied egg-rr 81.5%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in alpha around 0 67.0%

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

   9. Taylor expanded in beta around 0 66.4%

      \[\leadsto \frac{\frac{\beta + 1}{\color{blue}{6 + 5 \cdot \beta}}}{\alpha + \left(2 + \beta\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.4%

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

   11. Simplified 66.4%

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

   if 5.79999999999999982 < beta

   1. Initial program 78.8%

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

      1. associate-/l/ 75.5%

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

      2. associate-/r* 60.8%

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

      3. +-commutative 60.8%

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

      4. associate-+r+ 60.8%

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

      5. +-commutative 60.8%

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

      6. associate-+r+ 60.8%

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

      7. associate-+r+ 60.8%

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

      8. distribute-rgt1-in 60.8%

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

      9. +-commutative 60.8%

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

      10. *-commutative 60.8%

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

      11. distribute-rgt1-in 60.8%

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

      12. +-commutative 60.8%

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

      13. times-frac 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in beta around inf 81.6%

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

      1. associate-*l/ 81.8%

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

      2. +-commutative 81.8%

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

      3. +-commutative 81.8%

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

      4. associate-+l+ 81.8%

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

      5. +-commutative 81.8%

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

      6. associate-+l+ 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{\frac{1 + \beta}{6 + \beta \cdot 5}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 13: 97.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{\frac{1 + \beta}{6 + \beta \cdot 5}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6)
   (/ (/ (+ 1.0 beta) (+ 6.0 (* beta 5.0))) (+ alpha (+ 2.0 beta)))
   (/ (/ (- alpha -1.0) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = ((1.0d0 + beta) / (6.0d0 + (beta * 5.0d0))) / (alpha + (2.0d0 + beta))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta))
	else:
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(6.0 + Float64(beta * 5.0))) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = ((1.0 + beta) / (6.0 + (beta * 5.0))) / (alpha + (2.0 + beta));
	else
		tmp = ((alpha - -1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5999999999999996

   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/ 99.5%

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

      2. associate-/r* 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-+r+ 96.2%

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

      5. +-commutative 96.2%

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

      6. associate-+r+ 96.2%

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

      7. associate-+r+ 96.2%

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

      8. distribute-rgt1-in 96.2%

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

      9. +-commutative 96.2%

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

      10. *-commutative 96.2%

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

      11. distribute-rgt1-in 96.2%

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

      12. +-commutative 96.2%

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

      13. times-frac 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 81.5%

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

      3. *-commutative 81.5%

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

      4. +-commutative 81.5%

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

   5. Applied egg-rr 81.5%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r/ 99.5%

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

      6. times-frac 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in alpha around 0 67.0%

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

   9. Taylor expanded in beta around 0 66.4%

      \[\leadsto \frac{\frac{\beta + 1}{\color{blue}{6 + 5 \cdot \beta}}}{\alpha + \left(2 + \beta\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.4%

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

   11. Simplified 66.4%

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

   if 4.5999999999999996 < beta

   1. Initial program 78.8%

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

   2. Taylor expanded in beta around -inf 81.8%

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

      1. associate-*r/ 81.8%

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

      2. mul-1-neg 81.8%

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

      3. sub-neg 81.8%

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

      4. mul-1-neg 81.8%

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

      5. distribute-neg-in 81.8%

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

      6. +-commutative 81.8%

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

      7. mul-1-neg 81.8%

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

      8. distribute-lft-in 81.8%

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

      9. metadata-eval 81.8%

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

      10. mul-1-neg 81.8%

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

      11. unsub-neg 81.8%

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

   4. Simplified 81.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{\frac{1 + \beta}{6 + \beta \cdot 5}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 14: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2)
   (/
    (+ 0.16666666666666666 (* alpha 0.027777777777777776))
    (+ alpha (+ 2.0 beta)))
   (* (/ 1.0 beta) (/ (+ 1.0 alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.2d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.027777777777777776d0)) / (alpha + (2.0d0 + beta))
    else
        tmp = (1.0d0 / beta) * ((1.0d0 + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2:
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (2.0 + beta))
	else:
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.027777777777777776)) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(1.0 / beta) * 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 <= 8.2)
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (2.0 + beta));
	else
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.2], N[(N[(0.16666666666666666 + N[(alpha * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.1999999999999993

   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/ 99.5%

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

      2. associate-/r* 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-+r+ 96.2%

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

      5. +-commutative 96.2%

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

      6. associate-+r+ 96.2%

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

      7. associate-+r+ 96.2%

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

      8. distribute-rgt1-in 96.2%

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

      9. +-commutative 96.2%

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

      10. *-commutative 96.2%

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

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in beta around 0 97.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in alpha around 0 65.0%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \alpha}}{\alpha + \left(2 + \beta\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.0%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\alpha \cdot 0.027777777777777776}}{\alpha + \left(2 + \beta\right)} \]

   11. Simplified 65.0%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot 0.027777777777777776}}{\alpha + \left(2 + \beta\right)} \]

   if 8.1999999999999993 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 84.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \]

   5. Taylor expanded in beta around inf 81.4%

      \[\leadsto \frac{1 + \alpha}{\beta} \cdot \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \]

Alternative 15: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/
    (+ 0.16666666666666666 (* beta 0.027777777777777776))
    (+ alpha (+ 2.0 beta)))
   (* (/ 1.0 beta) (/ (+ 1.0 alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / (alpha + (2.0d0 + beta))
    else
        tmp = (1.0d0 / beta) * ((1.0d0 + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta))
	else:
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(1.0 / beta) * 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 <= 5.2)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta));
	else
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 66.4%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{\alpha + \left(2 + \beta\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.4%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{\alpha + \left(2 + \beta\right)} \]

   11. Simplified 66.4%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{\alpha + \left(2 + \beta\right)} \]

   if 5.20000000000000018 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 84.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \]

   5. Taylor expanded in beta around inf 81.4%

      \[\leadsto \frac{1 + \alpha}{\beta} \cdot \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \]

Alternative 16: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6)
   (/
    (+ 0.16666666666666666 (* beta 0.027777777777777776))
    (+ alpha (+ 2.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = (0.16666666666666666d0 + (beta * 0.027777777777777776d0)) / (alpha + (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = (0.16666666666666666 + (beta * 0.027777777777777776)) / (alpha + (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5999999999999996

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 66.4%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \beta}}{\alpha + \left(2 + \beta\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.4%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\beta \cdot 0.027777777777777776}}{\alpha + \left(2 + \beta\right)} \]

   11. Simplified 66.4%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \beta \cdot 0.027777777777777776}}{\alpha + \left(2 + \beta\right)} \]

   if 4.5999999999999996 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 81.6%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]
   5. Step-by-step derivation

      1. associate-*l/ 81.8%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 81.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)} \]

      3. +-commutative 81.8%

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\color{blue}{\left(\beta + 2\right) + \alpha}} \]

      4. associate-+l+ 81.8%

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\color{blue}{\beta + \left(2 + \alpha\right)}} \]

      5. +-commutative 81.8%

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\color{blue}{\left(2 + \alpha\right) + \beta}} \]

      6. associate-+l+ 81.8%

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\color{blue}{2 + \left(\alpha + \beta\right)}} \]

   6. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{2 + \left(\alpha + \beta\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 81.8%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot 1}{\beta}}}{2 + \left(\alpha + \beta\right)} \]

      2. *-rgt-identity 81.8%

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{2 + \left(\alpha + \beta\right)} \]

   8. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 17: 93.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5)
   (/ 0.16666666666666666 (+ alpha (+ 2.0 beta)))
   (* (+ 1.0 alpha) (/ 1.0 (* beta beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 + alpha) * (1.0 / (beta * beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.5d0) then
        tmp = 0.16666666666666666d0 / (alpha + (2.0d0 + beta))
    else
        tmp = (1.0d0 + alpha) * (1.0d0 / (beta * beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 + alpha) * (1.0 / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.5:
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta))
	else:
		tmp = (1.0 + alpha) * (1.0 / (beta * beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5)
		tmp = Float64(0.16666666666666666 / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(1.0 + alpha) * 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 <= 8.5)
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	else
		tmp = (1.0 + alpha) * (1.0 / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.5], N[(0.16666666666666666 / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 65.2%

      \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\alpha + \left(2 + \beta\right)} \]

   if 8.5 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.6%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   7. Step-by-step derivation

      1. div-inv 80.6%

         \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{1}{\beta \cdot \beta}} \]

   8. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\left(1 + \alpha\right) \cdot \frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 18: 96.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6)
   (/ 0.16666666666666666 (+ alpha (+ 2.0 beta)))
   (* (/ 1.0 beta) (/ (+ 1.0 alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.6d0) then
        tmp = 0.16666666666666666d0 / (alpha + (2.0d0 + beta))
    else
        tmp = (1.0d0 / beta) * ((1.0d0 + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta))
	else:
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(0.16666666666666666 / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(1.0 / beta) * 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 <= 7.6)
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	else
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.6], N[(0.16666666666666666 / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 65.2%

      \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\alpha + \left(2 + \beta\right)} \]

   if 7.5999999999999996 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 84.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta}} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \]

   5. Taylor expanded in beta around inf 81.4%

      \[\leadsto \frac{1 + \alpha}{\beta} \cdot \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \]

Alternative 19: 90.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6)
   (/ 0.16666666666666666 (+ alpha (+ 2.0 beta)))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} 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 <= 7.6d0) then
        tmp = 0.16666666666666666d0 / (alpha + (2.0d0 + beta))
    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 <= 7.6) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta))
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(0.16666666666666666 / Float64(alpha + Float64(2.0 + beta)));
	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 <= 7.6)
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	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, 7.6], N[(0.16666666666666666 / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 65.2%

      \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\alpha + \left(2 + \beta\right)} \]

   if 7.5999999999999996 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.6%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around 0 75.8%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 75.8%

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 20: 90.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.3:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.3)
   (/ 0.16666666666666666 (+ alpha (+ 2.0 beta)))
   (/ 1.0 (* beta (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.3) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.3d0) then
        tmp = 0.16666666666666666d0 / (alpha + (2.0d0 + beta))
    else
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.3) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.3:
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta))
	else:
		tmp = 1.0 / (beta * (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.3)
		tmp = Float64(0.16666666666666666 / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(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 <= 6.3)
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.3], N[(0.16666666666666666 / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.29999999999999982

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 65.2%

      \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\alpha + \left(2 + \beta\right)} \]

   if 6.29999999999999982 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 81.6%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]

   5. Taylor expanded in alpha around 0 75.8%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(\beta + 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.3:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \]

Alternative 21: 93.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5)
   (/ 0.16666666666666666 (+ alpha (+ 2.0 beta)))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 + 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 <= 8.5d0) then
        tmp = 0.16666666666666666d0 / (alpha + (2.0d0 + beta))
    else
        tmp = (1.0d0 + 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 <= 8.5) {
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.5:
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta))
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5)
		tmp = Float64(0.16666666666666666 / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(1.0 + alpha) / 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 <= 8.5)
		tmp = 0.16666666666666666 / (alpha + (2.0 + beta));
	else
		tmp = (1.0 + 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, 8.5], N[(0.16666666666666666 / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 65.2%

      \[\leadsto \frac{\color{blue}{0.16666666666666666}}{\alpha + \left(2 + \beta\right)} \]

   if 8.5 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.6%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 22: 47.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5)
   (+ (* alpha -0.041666666666666664) 0.08333333333333333)
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = (alpha * -0.041666666666666664) + 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.5d0) then
        tmp = (alpha * (-0.041666666666666664d0)) + 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = (alpha * -0.041666666666666664) + 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.5:
		tmp = (alpha * -0.041666666666666664) + 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5)
		tmp = Float64(Float64(alpha * -0.041666666666666664) + 0.08333333333333333);
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.5)
		tmp = (alpha * -0.041666666666666664) + 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.5], N[(N[(alpha * -0.041666666666666664), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 65.1%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   10. Taylor expanded in alpha around 0 63.2%

       \[\leadsto \color{blue}{-0.041666666666666664 \cdot \alpha + 0.08333333333333333} \]

   if 8.5 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 81.6%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]

   5. Taylor expanded in alpha around inf 6.7%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 23: 48.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.1e+107)
   (/ 0.16666666666666666 (+ 2.0 beta))
   (/ 1.0 (* alpha alpha))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.1e+107) {
		tmp = 0.16666666666666666 / (2.0 + beta);
	} else {
		tmp = 1.0 / (alpha * alpha);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.1d+107) then
        tmp = 0.16666666666666666d0 / (2.0d0 + beta)
    else
        tmp = 1.0d0 / (alpha * alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.1e+107) {
		tmp = 0.16666666666666666 / (2.0 + beta);
	} else {
		tmp = 1.0 / (alpha * alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 3.1e+107:
		tmp = 0.16666666666666666 / (2.0 + beta)
	else:
		tmp = 1.0 / (alpha * alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.1e+107)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + beta));
	else
		tmp = Float64(1.0 / Float64(alpha * alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.1e+107)
		tmp = 0.16666666666666666 / (2.0 + beta);
	else
		tmp = 1.0 / (alpha * alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 3.1e+107], N[(0.16666666666666666 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.10000000000000026e107

   1. Initial program 99.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 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-/r* 92.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 92.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 92.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 92.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 92.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 92.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 92.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 92.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 92.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 92.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 92.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 98.8%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 68.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 68.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 68.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 98.8%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 98.8%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 98.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 98.3%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in beta around 0 66.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in alpha around 0 54.1%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

   if 3.10000000000000026e107 < alpha

   1. Initial program 67.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 64.1%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 49.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 49.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 49.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 49.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 49.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 49.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 49.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 49.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 49.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 49.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 49.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 85.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 85.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 71.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 71.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 71.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 85.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 85.6%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 85.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 85.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 64.1%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in beta around 0 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in alpha around inf 80.7%

      \[\leadsto \color{blue}{\frac{1}{{\alpha}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 80.7%

          \[\leadsto \frac{1}{\color{blue}{\alpha \cdot \alpha}} \]

   11. Simplified 80.7%

       \[\leadsto \color{blue}{\frac{1}{\alpha \cdot \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 24: 90.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6) (/ 0.16666666666666666 (+ 2.0 beta)) (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.16666666666666666 / (2.0 + beta);
	} 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 <= 7.6d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + beta)
    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 <= 7.6) {
		tmp = 0.16666666666666666 / (2.0 + beta);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = 0.16666666666666666 / (2.0 + beta)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + beta));
	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 <= 7.6)
		tmp = 0.16666666666666666 / (2.0 + beta);
	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, 7.6], N[(0.16666666666666666 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   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/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in beta around 0 97.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in alpha around 0 64.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

   if 7.5999999999999996 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.6%

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   6. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   7. Taylor expanded in alpha around 0 75.8%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 75.8%

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 25: 47.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 96.2%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 96.2%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 96.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 96.2%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 96.2%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 96.2%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 99.5%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 81.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 81.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

      6. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

      9. associate-+r+ 99.8%

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

   8. Taylor expanded in alpha around 0 67.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

   9. Taylor expanded in beta around 0 65.1%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

   10. Taylor expanded in alpha around 0 64.1%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 12 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. associate-/r* 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

      3. +-commutative 60.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      4. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      5. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      6. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      7. associate-+r+ 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      8. distribute-rgt1-in 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      9. +-commutative 60.8%

         \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      10. *-commutative 60.8%

          \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      11. distribute-rgt1-in 60.8%

          \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      12. +-commutative 60.8%

          \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

      13. times-frac 89.6%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]

   4. Taylor expanded in beta around inf 81.6%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1}{\beta}} \]

   5. Taylor expanded in alpha around inf 6.7%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 26: 45.8% 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 92.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-/r* 83.2%

      \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

   3. +-commutative 83.2%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   4. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   5. +-commutative 83.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   6. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   7. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   8. distribute-rgt1-in 83.2%

      \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   9. +-commutative 83.2%

      \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   10. *-commutative 83.2%

       \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   11. distribute-rgt1-in 83.2%

       \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   12. +-commutative 83.2%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   13. times-frac 95.9%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 95.9%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 95.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

   2. expm1-udef 68.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

   3. *-commutative 68.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

   4. +-commutative 68.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

5. Applied egg-rr 68.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 95.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

   2. expm1-log1p 95.9%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

   3. associate-*r/ 95.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

   4. *-commutative 95.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

   5. associate-*r/ 90.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

   6. times-frac 99.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

   7. +-commutative 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

   8. +-commutative 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

   9. associate-+r+ 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

   10. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

8. Taylor expanded in alpha around 0 71.3%

   \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

9. Taylor expanded in beta around 0 43.3%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

10. Final simplification 43.3%

    \[\leadsto \frac{0.16666666666666666}{\alpha + 2} \]

Alternative 27: 47.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{2 + \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ 2.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (2.0 + beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (2.0d0 + beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (2.0 + beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (2.0 + beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(2.0 + beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (2.0 + beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-/r* 83.2%

      \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

   3. +-commutative 83.2%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   4. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   5. +-commutative 83.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   6. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   7. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   8. distribute-rgt1-in 83.2%

      \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   9. +-commutative 83.2%

      \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   10. *-commutative 83.2%

       \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   11. distribute-rgt1-in 83.2%

       \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   12. +-commutative 83.2%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   13. times-frac 95.9%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 95.9%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 95.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

   2. expm1-udef 68.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

   3. *-commutative 68.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

   4. +-commutative 68.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

5. Applied egg-rr 68.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 95.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

   2. expm1-log1p 95.9%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

   3. associate-*r/ 95.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

   4. *-commutative 95.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

   5. associate-*r/ 90.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

   6. times-frac 99.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

   7. +-commutative 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

   8. +-commutative 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

   9. associate-+r+ 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

   10. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

8. Taylor expanded in beta around 0 69.7%

   \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}}{\alpha + \left(2 + \beta\right)} \]

9. Taylor expanded in alpha around 0 43.1%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

10. Final simplification 43.1%

    \[\leadsto \frac{0.16666666666666666}{2 + \beta} \]

Alternative 28: 45.6% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 92.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. associate-/r* 83.2%

      \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]

   3. +-commutative 83.2%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   4. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   5. +-commutative 83.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   6. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \beta\right) + \alpha\right)} + \beta \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   7. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   8. distribute-rgt1-in 83.2%

      \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(\beta + 1\right) \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   9. +-commutative 83.2%

      \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\left(1 + \beta\right)} \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   10. *-commutative 83.2%

       \[\leadsto \frac{\left(1 + \beta\right) + \color{blue}{\alpha \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   11. distribute-rgt1-in 83.2%

       \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   12. +-commutative 83.2%

       \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(1 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]

   13. times-frac 95.9%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1 + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Simplified 95.9%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 95.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

   2. expm1-udef 68.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

   3. *-commutative 68.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

   4. +-commutative 68.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

5. Applied egg-rr 68.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 95.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

   2. expm1-log1p 95.9%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}} \]

   3. associate-*r/ 95.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \left(1 + \alpha\right)}{\alpha + \left(\beta + 2\right)}} \]

   4. *-commutative 95.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

   5. associate-*r/ 90.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{\alpha + \left(\beta + 2\right)} \]

   6. times-frac 99.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}}{\alpha + \left(\beta + 2\right)} \]

   7. +-commutative 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

   8. +-commutative 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\beta + \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

   9. associate-+r+ 99.8%

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\beta + \alpha\right) + 3}}}{\alpha + \left(\beta + 2\right)} \]

   10. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right)} + 3}}{\alpha + \left(\beta + 2\right)} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \color{blue}{\left(2 + \beta\right)}} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \beta\right) + 3}}{\alpha + \left(2 + \beta\right)}} \]

8. Taylor expanded in alpha around 0 71.3%

   \[\leadsto \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}}{\alpha + \left(2 + \beta\right)} \]

9. Taylor expanded in beta around 0 43.3%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \alpha}} \]

10. Taylor expanded in alpha around 0 42.1%

    \[\leadsto \color{blue}{0.08333333333333333} \]

11. Final simplification 42.1%

    \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023274 
(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)))