Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 98.6%

Time: 20.1s

Alternatives: 15

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 2}}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.5e+16)
   (/ (/ (/ (+ 1.0 beta) (+ beta 2.0)) (+ 2.0 (+ alpha beta))) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5e+16) {
		tmp = (((1.0 + beta) / (beta + 2.0)) / (2.0 + (alpha + beta))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.5d+16) then
        tmp = (((1.0d0 + beta) / (beta + 2.0d0)) / (2.0d0 + (alpha + beta))) / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5e+16) {
		tmp = (((1.0 + beta) / (beta + 2.0)) / (2.0 + (alpha + beta))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.5e+16:
		tmp = (((1.0 + beta) / (beta + 2.0)) / (2.0 + (alpha + beta))) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.5e+16)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(2.0 + Float64(alpha + beta))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.5e+16)
		tmp = (((1.0 + beta) / (beta + 2.0)) / (2.0 + (alpha + beta))) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.5e+16], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5e16

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0 68.1%

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

   4. Taylor expanded in alpha around 0 68.2%

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

      1. +-commutative 68.2%

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

   6. Simplified 68.2%

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

   if 7.5e16 < beta

   1. Initial program 84.1%

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

   3. Taylor expanded in alpha around 0 78.3%

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

   4. Taylor expanded in beta around inf 88.1%

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

   5. Taylor expanded in beta around inf 88.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 2}}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.8)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ 1.0 (* (+ alpha 2.0) (+ alpha 3.0))))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.8d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (1.0d0 / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.8:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(1.0 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.8)
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.79999999999999982

   1. Initial program 99.9%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in beta around 0 96.2%

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

   if 6.79999999999999982 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 87.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \alpha}{\frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{\frac{1 + \beta}{\beta + 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+38)
   (/
    (+ 1.0 alpha)
    (/ (* (+ beta 2.0) (+ beta 3.0)) (/ (+ 1.0 beta) (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+38) {
		tmp = (1.0 + alpha) / (((beta + 2.0) * (beta + 3.0)) / ((1.0 + beta) / (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+38) then
        tmp = (1.0d0 + alpha) / (((beta + 2.0d0) * (beta + 3.0d0)) / ((1.0d0 + beta) / (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+38) {
		tmp = (1.0 + alpha) / (((beta + 2.0) * (beta + 3.0)) / ((1.0 + beta) / (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5e+38:
		tmp = (1.0 + alpha) / (((beta + 2.0) * (beta + 3.0)) / ((1.0 + beta) / (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+38)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(Float64(beta + 2.0) * Float64(beta + 3.0)) / Float64(Float64(1.0 + beta) / Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+38)
		tmp = (1.0 + alpha) / (((beta + 2.0) * (beta + 3.0)) / ((1.0 + beta) / (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5e+38], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999997e38

   1. Initial program 99.8%

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

   3. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.8%

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

      3. frac-times 94.3%

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

      4. *-un-lft-identity 94.3%

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

      5. +-commutative 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. associate-*l/ 94.3%

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

   8. Applied egg-rr 94.3%

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

      1. associate-/l* 94.3%

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

      2. +-commutative 94.3%

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

      3. +-commutative 94.3%

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

      4. +-commutative 94.3%

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

      5. +-commutative 94.3%

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

      6. +-commutative 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in alpha around 0 84.2%

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

       1. +-commutative 84.2%

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

   13. Simplified 84.2%

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

   14. Taylor expanded in alpha around 0 66.9%

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

       1. +-commutative 66.9%

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

       2. +-commutative 66.9%

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

   16. Simplified 66.9%

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

   if 4.9999999999999997e38 < beta

   1. Initial program 82.6%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 89.7%

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

   5. Taylor expanded in beta around inf 89.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \alpha}{\frac{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}{\frac{1 + \beta}{\beta + 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6e+38)
   (/
    (/ (+ 1.0 beta) (+ beta 2.0))
    (* (+ 3.0 (+ alpha beta)) (+ alpha (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+38) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (alpha + beta)) * (alpha + (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6d+38) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((3.0d0 + (alpha + beta)) * (alpha + (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+38) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (alpha + beta)) * (alpha + (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6e+38:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (alpha + beta)) * (alpha + (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6e+38)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(3.0 + Float64(alpha + beta)) * Float64(alpha + Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6e+38)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((3.0 + (alpha + beta)) * (alpha + (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6e+38], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.0000000000000002e38

   1. Initial program 99.8%

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

      1. associate-/l/ 98.8%

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

      2. +-commutative 98.8%

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

      3. associate-+l+ 98.8%

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

      4. *-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-+l+ 98.8%

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

      7. metadata-eval 98.8%

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

      8. +-commutative 98.8%

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

      9. metadata-eval 98.8%

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

      10. metadata-eval 98.8%

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

      11. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in alpha around 0 85.8%

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

   if 6.0000000000000002e38 < beta

   1. Initial program 82.6%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 89.7%

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

   5. Taylor expanded in beta around inf 89.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.9%

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

3. Simplified 97.1%

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

   1. *-commutative 97.1%

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

   2. clear-num 97.1%

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

   3. frac-times 93.2%

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

   4. *-un-lft-identity 93.2%

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

   5. +-commutative 93.2%

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

6. Applied egg-rr 93.2%

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

   1. associate-*l/ 88.1%

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

8. Applied egg-rr 88.1%

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

   1. associate-/l* 93.2%

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

   2. +-commutative 93.2%

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

   3. +-commutative 93.2%

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

   4. +-commutative 93.2%

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

   5. +-commutative 93.2%

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

   6. +-commutative 93.2%

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

10. Simplified 93.2%

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

11. Taylor expanded in alpha around 0 86.0%

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

    1. +-commutative 86.0%

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

13. Simplified 86.0%

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

    1. expm1-log1p-u 86.0%

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

    2. expm1-udef 73.6%

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

    3. associate-/r/ 73.6%

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

    4. *-commutative 73.6%

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

    5. +-commutative 73.6%

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

    6. +-commutative 73.6%

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

    7. +-commutative 73.6%

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

    8. +-commutative 73.6%

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

    9. +-commutative 73.6%

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

15. Applied egg-rr 73.6%

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

    1. expm1-def 86.0%

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

    2. expm1-log1p 86.0%

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

    3. associate-/r* 74.5%

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

    4. +-commutative 74.5%

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

    5. +-commutative 74.5%

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

    6. +-commutative 74.5%

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

    7. +-commutative 74.5%

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

    8. +-commutative 74.5%

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

    9. +-commutative 74.5%

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

17. Simplified 74.5%

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

18. Final simplification 74.5%

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

Alternative 6: 97.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1.3500000000000001

   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 98.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

   6. Taylor expanded in beta around 0 96.2%

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

   if 1.3500000000000001 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 87.7%

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

4. Final simplification 93.4%

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

Alternative 7: 97.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 99.9%

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

   3. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-times 94.6%

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

      4. *-un-lft-identity 94.6%

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

      5. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in beta around 0 92.8%

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

   if 3.7999999999999998 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 87.7%

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

4. Final simplification 91.1%

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

Alternative 8: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7)
   (/ (+ 1.0 alpha) (* (+ alpha (+ beta 2.0)) (- 6.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.7d0) then
        tmp = (1.0d0 + alpha) / ((alpha + (beta + 2.0d0)) * (6.0d0 - beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(6.0 - beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7)
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.7], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(6.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7000000000000002

   1. Initial program 99.9%

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

   3. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-times 94.6%

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

      4. *-un-lft-identity 94.6%

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

      5. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in alpha around 0 66.8%

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

   8. Taylor expanded in beta around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. unsub-neg 66.5%

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

   10. Simplified 66.5%

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

   if 2.7000000000000002 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 87.7%

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

   5. Taylor expanded in beta around inf 87.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (/ (+ 1.0 alpha) (* (+ alpha (+ beta 2.0)) (- 6.0 beta)))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d0) then
        tmp = (1.0d0 + alpha) / ((alpha + (beta + 2.0d0)) * (6.0d0 - beta))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(6.0 - beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = (1.0 + alpha) / ((alpha + (beta + 2.0)) * (6.0 - beta));
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(6.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   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 98.8%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-times 94.6%

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

      4. *-un-lft-identity 94.6%

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

      5. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in alpha around 0 66.8%

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

   8. Taylor expanded in beta around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. unsub-neg 66.5%

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

   10. Simplified 66.5%

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

   if 1.5 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 87.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6)
   (/ 0.16666666666666666 (+ beta 2.0))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.6d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.6)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.6], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   1. Initial program 99.9%

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

   3. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-times 94.6%

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

      4. *-un-lft-identity 94.6%

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

      5. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in beta around 0 92.8%

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

   8. Taylor expanded in alpha around 0 66.0%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 66.0%

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

   10. Simplified 66.0%

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

   if 7.5999999999999996 < beta

   1. Initial program 84.4%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in beta around inf 87.4%

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

   6. Taylor expanded in beta around inf 87.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   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 98.8%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-times 94.6%

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

      4. *-un-lft-identity 94.6%

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

      5. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in beta around 0 92.8%

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

   8. Taylor expanded in alpha around 0 66.0%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 66.0%

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

   10. Simplified 66.0%

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

   if 5.4000000000000004 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0 78.8%

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

   4. Taylor expanded in beta around inf 87.7%

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

   5. Taylor expanded in beta around inf 87.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   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 98.8%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-times 94.6%

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

      4. *-un-lft-identity 94.6%

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

      5. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in beta around 0 92.8%

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

   8. Taylor expanded in alpha around 0 66.0%

      \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
   9. Step-by-step derivation

      1. +-commutative 66.0%

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

   10. Simplified 66.0%

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

   if 5.4000000000000004 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around -inf 87.6%

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

   4. Taylor expanded in alpha around 0 82.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) 0.08333333333333333 (/ 0.5 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.5 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.5d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.5 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.5 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.5 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.5 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], 0.08333333333333333, N[(0.5 / 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 98.8%

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

   5. Taylor expanded in beta around 0 97.3%

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

   6. Taylor expanded in alpha around 0 65.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 65.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(1 + \beta\right)}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

      2. times-frac 65.9%

         \[\leadsto \color{blue}{\frac{0.5}{2 + \beta} \cdot \frac{1 + \beta}{3 + \beta}} \]

      3. +-commutative 65.9%

         \[\leadsto \frac{0.5}{2 + \beta} \cdot \frac{\color{blue}{\beta + 1}}{3 + \beta} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{0.5}{2 + \beta} \cdot \frac{\beta + 1}{\color{blue}{\beta + 3}} \]

   8. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{0.5}{2 + \beta} \cdot \frac{\beta + 1}{\beta + 3}} \]

   9. Taylor expanded in beta around 0 65.9%

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

   if 6 < beta

   1. Initial program 84.4%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in beta around 0 61.1%

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

   6. Taylor expanded in alpha around 0 51.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 51.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(1 + \beta\right)}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

      2. times-frac 6.5%

         \[\leadsto \color{blue}{\frac{0.5}{2 + \beta} \cdot \frac{1 + \beta}{3 + \beta}} \]

      3. +-commutative 6.5%

         \[\leadsto \frac{0.5}{2 + \beta} \cdot \frac{\color{blue}{\beta + 1}}{3 + \beta} \]

      4. +-commutative 6.5%

         \[\leadsto \frac{0.5}{2 + \beta} \cdot \frac{\beta + 1}{\color{blue}{\beta + 3}} \]

   8. Simplified 6.5%

      \[\leadsto \color{blue}{\frac{0.5}{2 + \beta} \cdot \frac{\beta + 1}{\beta + 3}} \]

   9. Taylor expanded in beta around inf 6.5%

      \[\leadsto \color{blue}{\frac{0.5}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\beta + 2} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (beta + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (beta + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(beta + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (beta + 2.0);
end

Wolfram

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

Derivation

1. Initial program 94.9%

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

3. Simplified 97.1%

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

   1. *-commutative 97.1%

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

   2. clear-num 97.1%

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

   3. frac-times 93.2%

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

   4. *-un-lft-identity 93.2%

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

   5. +-commutative 93.2%

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

6. Applied egg-rr 93.2%

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

7. Taylor expanded in beta around 0 69.3%

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

8. Taylor expanded in alpha around 0 46.7%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
9. Step-by-step derivation

   1. +-commutative 46.7%

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

10. Simplified 46.7%

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

11. Final simplification 46.7%

    \[\leadsto \frac{0.16666666666666666}{\beta + 2} \]
12. Add Preprocessing

Alternative 15: 44.6% 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 94.9%

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

3. Simplified 97.1%

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

5. Taylor expanded in beta around 0 85.6%

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

6. Taylor expanded in alpha around 0 61.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 61.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(1 + \beta\right)}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   2. times-frac 46.6%

      \[\leadsto \color{blue}{\frac{0.5}{2 + \beta} \cdot \frac{1 + \beta}{3 + \beta}} \]

   3. +-commutative 46.6%

      \[\leadsto \frac{0.5}{2 + \beta} \cdot \frac{\color{blue}{\beta + 1}}{3 + \beta} \]

   4. +-commutative 46.6%

      \[\leadsto \frac{0.5}{2 + \beta} \cdot \frac{\beta + 1}{\color{blue}{\beta + 3}} \]

8. Simplified 46.6%

   \[\leadsto \color{blue}{\frac{0.5}{2 + \beta} \cdot \frac{\beta + 1}{\beta + 3}} \]

9. Taylor expanded in beta around 0 45.8%

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

10. Final simplification 45.8%

    \[\leadsto 0.08333333333333333 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))