Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.8%

Time: 26.6s

Alternatives: 16

Speedup: 2.9×

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: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 / (t_0 / ((1.0d0 + beta) / (alpha + (beta + 3.0d0))))) / (t_0 / (1.0d0 + alpha))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.0%

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

3. Simplified 98.5%

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

   1. *-commutative 98.5%

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

   2. clear-num 98.4%

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

   3. clear-num 98.4%

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

   4. frac-times 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-/r* 98.4%

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

   2. associate-/l* 99.5%

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

   3. +-commutative 99.5%

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

   4. +-commutative 99.5%

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

   5. +-commutative 99.5%

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

   6. +-commutative 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Simplified 98.5%

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

   1. clear-num 98.5%

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

   2. associate-+r+ 98.5%

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

   3. *-commutative 98.5%

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

   4. frac-times 94.9%

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

   5. *-un-lft-identity 94.9%

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

   6. +-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-+r+ 94.9%

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

6. Applied egg-rr 94.9%

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

   1. associate-/r* 98.5%

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

   2. associate-/l* 95.7%

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

   3. associate-*l/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*r/ 95.7%

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

   6. associate-*l/ 98.5%

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

   7. associate-/l* 98.5%

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

   8. +-commutative 98.5%

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

   9. associate-/l* 99.5%

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

   10. +-commutative 99.5%

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

   11. +-commutative 99.5%

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

   12. +-commutative 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 50000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{2 + \left(\beta - \alpha \cdot \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 50000000.0)
   (/
    (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 2.0 beta)))
    (+ 2.0 (- beta (* alpha beta))))
   (/
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ (+ beta 4.0) (* alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 50000000.0) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + (beta - (alpha * beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 50000000.0d0) then
        tmp = ((1.0d0 + beta) / ((beta + 3.0d0) * (2.0d0 + beta))) / (2.0d0 + (beta - (alpha * beta)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / ((beta + 4.0d0) + (alpha * 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 50000000.0) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + (beta - (alpha * beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 50000000.0:
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + (beta - (alpha * beta)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 50000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))) / Float64(2.0 + Float64(beta - Float64(alpha * beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 50000000.0)
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + (beta - (alpha * beta)));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 50000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta - N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e7

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in alpha around 0 68.7%

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

       1. mul-1-neg 68.7%

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

       2. unsub-neg 68.7%

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

       3. *-commutative 68.7%

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

   14. Simplified 68.7%

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

   15. Taylor expanded in beta around inf 68.6%

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

   if 5e7 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

      1. clear-num 96.0%

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

      2. associate-+r+ 96.0%

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

      3. *-commutative 96.0%

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

      4. frac-times 83.3%

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

      5. *-un-lft-identity 83.3%

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

      6. +-commutative 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-+r+ 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-/r* 96.0%

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

      2. associate-/l* 86.1%

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

      3. associate-*l/ 96.0%

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

      4. *-commutative 96.0%

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

      5. associate-*r/ 86.1%

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

      6. associate-*l/ 95.9%

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

      7. associate-/l* 95.9%

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

      8. +-commutative 95.9%

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

      9. associate-/l* 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 83.5%

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

       1. associate-+r+ 83.5%

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

   11. Simplified 83.5%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\left(4 + \beta\right) + 2 \cdot \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 50000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{2 + \left(\beta - \alpha \cdot \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 410000000:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 410000000.0)
   (/ (/ (/ (+ 1.0 beta) (+ beta 3.0)) (+ 2.0 beta)) (+ 2.0 beta))
   (/
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ (+ beta 4.0) (* alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 410000000.0) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 410000000.0d0) then
        tmp = (((1.0d0 + beta) / (beta + 3.0d0)) / (2.0d0 + beta)) / (2.0d0 + beta)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / ((beta + 4.0d0) + (alpha * 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 410000000.0) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 410000000.0:
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta)
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 410000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(beta + 3.0)) / Float64(2.0 + beta)) / Float64(2.0 + beta));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 410000000.0)
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta);
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / ((beta + 4.0) + (alpha * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 410000000.0], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.1e8

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

       1. *-un-lft-identity 68.3%

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

       2. times-frac 68.3%

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

       3. +-commutative 68.3%

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

   13. Applied egg-rr 68.3%

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

       1. associate-*l/ 68.3%

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

       2. +-commutative 68.3%

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

       3. *-lft-identity 68.3%

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

       4. +-commutative 68.3%

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

       5. +-commutative 68.3%

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

   15. Simplified 68.3%

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

   16. Taylor expanded in alpha around 0 68.4%

       \[\leadsto \frac{\frac{\frac{1 + \beta}{\beta + 3}}{\beta + 2}}{\color{blue}{2 + \beta}} \]
   17. Step-by-step derivation

       1. +-commutative 68.4%

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

   18. Simplified 68.4%

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

   if 4.1e8 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

      1. clear-num 96.0%

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

      2. associate-+r+ 96.0%

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

      3. *-commutative 96.0%

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

      4. frac-times 83.3%

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

      5. *-un-lft-identity 83.3%

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

      6. +-commutative 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-+r+ 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-/r* 96.0%

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

      2. associate-/l* 86.1%

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

      3. associate-*l/ 96.0%

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

      4. *-commutative 96.0%

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

      5. associate-*r/ 86.1%

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

      6. associate-*l/ 95.9%

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

      7. associate-/l* 95.9%

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

      8. +-commutative 95.9%

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

      9. associate-/l* 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 83.5%

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

       1. associate-+r+ 83.5%

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

   11. Simplified 83.5%

       \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\left(4 + \beta\right) + 2 \cdot \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 410000000:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Simplified 98.5%

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

5. Taylor expanded in alpha around 0 72.9%

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

6. Final simplification 72.9%

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

Alternative 6: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Simplified 98.5%

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

   1. *-commutative 98.5%

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

   2. clear-num 98.4%

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

   3. clear-num 98.4%

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

   4. frac-times 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-/r* 98.4%

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

   2. associate-/l* 99.5%

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

   3. +-commutative 99.5%

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

   4. +-commutative 99.5%

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

   5. +-commutative 99.5%

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

   6. +-commutative 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in alpha around 0 72.9%

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

    1. +-commutative 72.9%

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

    2. +-commutative 72.9%

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

11. Simplified 72.9%

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

    1. *-un-lft-identity 72.9%

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

    2. times-frac 72.5%

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

    3. +-commutative 72.5%

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

13. Applied egg-rr 72.5%

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

    1. associate-*l/ 72.5%

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

    2. +-commutative 72.5%

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

    3. *-lft-identity 72.5%

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

    4. +-commutative 72.5%

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

    5. +-commutative 72.5%

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

15. Simplified 72.5%

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

16. Final simplification 72.5%

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

Alternative 7: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8e+17)
   (/ (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 2.0 beta))) (+ 2.0 beta))
   (/ (/ 1.0 beta) (/ (+ alpha (+ 2.0 beta)) (+ 1.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+17) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta);
	} else {
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.8d+17) then
        tmp = ((1.0d0 + beta) / ((beta + 3.0d0) * (2.0d0 + beta))) / (2.0d0 + beta)
    else
        tmp = (1.0d0 / beta) / ((alpha + (2.0d0 + beta)) / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+17) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta);
	} else {
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8e+17:
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta)
	else:
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8e+17)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))) / Float64(2.0 + beta));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(Float64(alpha + Float64(2.0 + beta)) / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8e+17)
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta);
	else
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8e+17], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8e17

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in alpha around 0 68.4%

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

       1. +-commutative 68.4%

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

   14. Simplified 68.4%

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

   if 3.8e17 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. clear-num 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

      4. frac-times 83.0%

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

      5. metadata-eval 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-/r* 82.9%

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

      3. +-commutative 82.9%

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

      4. +-commutative 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.8e+16)
   (/ (/ (/ (+ 1.0 beta) (+ beta 3.0)) (+ 2.0 beta)) (+ 2.0 beta))
   (/ (/ 1.0 beta) (/ (+ alpha (+ 2.0 beta)) (+ 1.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8e+16) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta);
	} else {
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.8d+16) then
        tmp = (((1.0d0 + beta) / (beta + 3.0d0)) / (2.0d0 + beta)) / (2.0d0 + beta)
    else
        tmp = (1.0d0 / beta) / ((alpha + (2.0d0 + beta)) / (1.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8e+16) {
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta);
	} else {
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.8e+16:
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta)
	else:
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.8e+16)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(beta + 3.0)) / Float64(2.0 + beta)) / Float64(2.0 + beta));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(Float64(alpha + Float64(2.0 + beta)) / Float64(1.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.8e+16)
		tmp = (((1.0 + beta) / (beta + 3.0)) / (2.0 + beta)) / (2.0 + beta);
	else
		tmp = (1.0 / beta) / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.8e+16], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.8e16

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

       1. *-un-lft-identity 68.3%

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

       2. times-frac 68.3%

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

       3. +-commutative 68.3%

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

   13. Applied egg-rr 68.3%

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

       1. associate-*l/ 68.3%

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

       2. +-commutative 68.3%

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

       3. *-lft-identity 68.3%

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

       4. +-commutative 68.3%

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

       5. +-commutative 68.3%

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

   15. Simplified 68.3%

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

   16. Taylor expanded in alpha around 0 68.4%

       \[\leadsto \frac{\frac{\frac{1 + \beta}{\beta + 3}}{\beta + 2}}{\color{blue}{2 + \beta}} \]
   17. Step-by-step derivation

       1. +-commutative 68.4%

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

   18. Simplified 68.4%

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

   if 5.8e16 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. clear-num 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

      4. frac-times 83.0%

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

      5. metadata-eval 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-/r* 82.9%

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

      3. +-commutative 82.9%

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

      4. +-commutative 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 3}}{2 + \beta}}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}\\ \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ alpha (+ 2.0 beta)) (+ 1.0 alpha))))
   (if (<= beta 6.0) (/ 0.16666666666666666 t_0) (/ (/ 1.0 beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + (2.0 + beta)) / (1.0 + alpha);
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / t_0;
	} else {
		tmp = (1.0 / 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)) / (1.0d0 + alpha)
    if (beta <= 6.0d0) then
        tmp = 0.16666666666666666d0 / t_0
    else
        tmp = (1.0d0 / 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)) / (1.0 + alpha);
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / t_0;
	} else {
		tmp = (1.0 / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + (2.0 + beta)) / (1.0 + alpha)
	tmp = 0
	if beta <= 6.0:
		tmp = 0.16666666666666666 / t_0
	else:
		tmp = (1.0 / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + Float64(2.0 + beta)) / Float64(1.0 + alpha))
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.16666666666666666 / t_0);
	else
		tmp = Float64(Float64(1.0 / 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)) / (1.0 + alpha);
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666 / t_0;
	else
		tmp = (1.0 / 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[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.0], N[(0.16666666666666666 / t$95$0), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in beta around 0 67.8%

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

   if 6 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. clear-num 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

      4. frac-times 83.0%

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

      5. metadata-eval 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-/r* 82.9%

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

      3. +-commutative 82.9%

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

      4. +-commutative 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.5)
   (/ 0.16666666666666666 (/ (+ alpha (+ 2.0 beta)) (+ 1.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = 0.16666666666666666 / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	} 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 <= 7.5d0) then
        tmp = 0.16666666666666666d0 / ((alpha + (2.0d0 + beta)) / (1.0d0 + alpha))
    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 <= 7.5) {
		tmp = 0.16666666666666666 / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.5:
		tmp = 0.16666666666666666 / ((alpha + (2.0 + beta)) / (1.0 + alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.5)
		tmp = Float64(0.16666666666666666 / Float64(Float64(alpha + Float64(2.0 + beta)) / Float64(1.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.5)
		tmp = 0.16666666666666666 / ((alpha + (2.0 + beta)) / (1.0 + alpha));
	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, 7.5], N[(0.16666666666666666 / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in beta around 0 67.8%

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

   if 7.5 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. un-div-inv 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

   7. Applied egg-rr 82.9%

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

   8. Taylor expanded in beta around inf 82.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{t_0}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 6.0)
     (/ 0.16666666666666666 (/ t_0 (+ 1.0 alpha)))
     (/ (/ (+ 1.0 alpha) t_0) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666 / (t_0 / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / t_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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 6.0d0) then
        tmp = 0.16666666666666666d0 / (t_0 / (1.0d0 + alpha))
    else
        tmp = ((1.0d0 + alpha) / t_0) / 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 <= 6.0) {
		tmp = 0.16666666666666666 / (t_0 / (1.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / t_0) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 6.0:
		tmp = 0.16666666666666666 / (t_0 / (1.0 + alpha))
	else:
		tmp = ((1.0 + alpha) / t_0) / 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 <= 6.0)
		tmp = Float64(0.16666666666666666 / Float64(t_0 / Float64(1.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / 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 <= 6.0)
		tmp = 0.16666666666666666 / (t_0 / (1.0 + alpha));
	else
		tmp = ((1.0 + alpha) / t_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_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.0], N[(0.16666666666666666 / N[(t$95$0 / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in beta around 0 67.8%

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

   if 6 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. un-div-inv 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

   7. Applied egg-rr 82.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{\frac{\alpha + \left(2 + \beta\right)}{1 + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(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 <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in beta around 0 67.8%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   13. Step-by-step derivation

       1. *-commutative 67.8%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   14. Simplified 67.8%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   15. Taylor expanded in alpha around 0 67.9%

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

   if 2.60000000000000009 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

   6. Taylor expanded in alpha around 0 78.8%

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

4. Final simplification 71.0%

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

Alternative 13: 91.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(1.0 / 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 <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = (1.0 / beta) / (2.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in beta around 0 67.8%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   13. Step-by-step derivation

       1. *-commutative 67.8%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   14. Simplified 67.8%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   15. Taylor expanded in alpha around 0 67.9%

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

   if 2.60000000000000009 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

   6. Taylor expanded in alpha around 0 78.8%

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

      1. associate-/r* 78.8%

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

      2. +-commutative 78.8%

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

   8. Simplified 78.8%

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

4. Final simplification 71.0%

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

Alternative 14: 96.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5) 0.08333333333333333 (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 0.08333333333333333;
	} 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 <= 3.5d0) then
        tmp = 0.08333333333333333d0
    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 <= 3.5) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.5:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.5)
		tmp = 0.08333333333333333;
	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, 3.5], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in beta around 0 67.8%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   13. Step-by-step derivation

       1. *-commutative 67.8%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   14. Simplified 67.8%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   15. Taylor expanded in alpha around 0 67.9%

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

   if 3.5 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. un-div-inv 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

   7. Applied egg-rr 82.9%

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

   8. Taylor expanded in beta around inf 82.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.3% accurate, 4.4× 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%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in alpha around 0 68.3%

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

       1. +-commutative 68.3%

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

       2. +-commutative 68.3%

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

   11. Simplified 68.3%

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

   12. Taylor expanded in beta around 0 67.8%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   13. Step-by-step derivation

       1. *-commutative 67.8%

          \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   14. Simplified 67.8%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

   15. Taylor expanded in alpha around 0 67.9%

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

   if 12 < beta

   1. Initial program 86.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in beta around inf 82.9%

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

      1. un-div-inv 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

   7. Applied egg-rr 82.9%

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

   8. Taylor expanded in alpha around inf 6.8%

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

4. Final simplification 50.5%

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

Alternative 16: 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 96.0%

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

3. Simplified 98.5%

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

   1. *-commutative 98.5%

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

   2. clear-num 98.4%

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

   3. clear-num 98.4%

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

   4. frac-times 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-/r* 98.4%

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

   2. associate-/l* 99.5%

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

   3. +-commutative 99.5%

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

   4. +-commutative 99.5%

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

   5. +-commutative 99.5%

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

   6. +-commutative 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in alpha around 0 72.9%

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

    1. +-commutative 72.9%

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

    2. +-commutative 72.9%

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

11. Simplified 72.9%

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

12. Taylor expanded in beta around 0 49.6%

    \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
13. Step-by-step derivation

    1. *-commutative 49.6%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

14. Simplified 49.6%

    \[\leadsto \color{blue}{\frac{1 + \alpha}{2 + \alpha} \cdot 0.16666666666666666} \]

15. Taylor expanded in alpha around 0 49.6%

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

16. Final simplification 49.6%

    \[\leadsto 0.08333333333333333 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024022 
(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)))