Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 20.5s

Alternatives: 16

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: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := 2 + \left(\beta + \alpha\right)\\ t_2 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \alpha}{t_2} \cdot \frac{1 + \beta}{t_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{t_1} \cdot \frac{\frac{1 + \alpha}{t_0}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0)))
        (t_1 (+ 2.0 (+ beta alpha)))
        (t_2 (+ alpha (+ beta 2.0))))
   (if (<= beta 8e+15)
     (* (/ (+ 1.0 alpha) t_2) (/ (+ 1.0 beta) (* t_2 t_0)))
     (* (/ beta t_1) (/ (/ (+ 1.0 alpha) t_0) t_1)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = 2.0 + (beta + alpha);
	double t_2 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 8e+15) {
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_2 * t_0));
	} else {
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = 2.0d0 + (beta + alpha)
    t_2 = alpha + (beta + 2.0d0)
    if (beta <= 8d+15) then
        tmp = ((1.0d0 + alpha) / t_2) * ((1.0d0 + beta) / (t_2 * t_0))
    else
        tmp = (beta / t_1) * (((1.0d0 + alpha) / t_0) / t_1)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = 2.0 + (beta + alpha);
	double t_2 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 8e+15) {
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_2 * t_0));
	} else {
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = 2.0 + (beta + alpha)
	t_2 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 8e+15:
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_2 * t_0))
	else:
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(2.0 + Float64(beta + alpha))
	t_2 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 8e+15)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_2) * Float64(Float64(1.0 + beta) / Float64(t_2 * t_0)));
	else
		tmp = Float64(Float64(beta / t_1) * Float64(Float64(Float64(1.0 + alpha) / t_0) / t_1));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = 2.0 + (beta + alpha);
	t_2 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 8e+15)
		tmp = ((1.0 + alpha) / t_2) * ((1.0 + beta) / (t_2 * t_0));
	else
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8e+15], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta / t$95$1), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8e15

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

   if 8e15 < beta

   1. Initial program 81.3%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in beta around inf 61.8%

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

      1. times-frac 90.0%

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

      2. associate-+r+ 90.0%

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

      3. +-commutative 90.0%

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

      4. +-commutative 90.0%

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

      5. *-commutative 90.0%

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

      6. +-commutative 90.0%

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

      7. associate-+r+ 90.0%

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

      8. associate-+r+ 90.0%

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

      9. +-commutative 90.0%

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

      10. +-commutative 90.0%

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

   7. Applied egg-rr 90.0%

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

      1. +-commutative 90.0%

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

      2. associate-/r* 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 60000000000000:\\ \;\;\;\;\frac{\frac{1 + \left(\beta + \alpha\right)}{t_1}}{t_1 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{t_0} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 60000000000000.0)
     (/ (/ (+ 1.0 (+ beta alpha)) t_1) (* t_1 (+ 3.0 (+ beta alpha))))
     (* (/ beta t_0) (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 60000000000000.0) {
		tmp = ((1.0 + (beta + alpha)) / t_1) / (t_1 * (3.0 + (beta + alpha)));
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 60000000000000.0d0) then
        tmp = ((1.0d0 + (beta + alpha)) / t_1) / (t_1 * (3.0d0 + (beta + alpha)))
    else
        tmp = (beta / t_0) * (((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 60000000000000.0) {
		tmp = ((1.0 + (beta + alpha)) / t_1) / (t_1 * (3.0 + (beta + alpha)));
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 60000000000000.0:
		tmp = ((1.0 + (beta + alpha)) / t_1) / (t_1 * (3.0 + (beta + alpha)))
	else:
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 60000000000000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(beta + alpha)) / t_1) / Float64(t_1 * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(beta / t_0) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 60000000000000.0)
		tmp = ((1.0 + (beta + alpha)) / t_1) / (t_1 * (3.0 + (beta + alpha)));
	else
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 60000000000000.0], N[(N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6e13

   1. Initial program 99.9%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in alpha around 0 99.5%

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

   if 6e13 < beta

   1. Initial program 81.6%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in beta around inf 62.3%

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

      1. times-frac 90.1%

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

      2. associate-+r+ 90.1%

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

      3. +-commutative 90.1%

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

      4. +-commutative 90.1%

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

      5. *-commutative 90.1%

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

      6. +-commutative 90.1%

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

      7. associate-+r+ 90.1%

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

      8. associate-+r+ 90.1%

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

      9. +-commutative 90.1%

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

      10. +-commutative 90.1%

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

   7. Applied egg-rr 90.1%

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

      1. +-commutative 90.1%

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

      2. associate-/r* 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 60000000000000:\\ \;\;\;\;\frac{\frac{1 + \left(\beta + \alpha\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / t$95$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. clear-num 97.1%

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

   2. associate-+r+ 97.1%

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

   3. *-commutative 97.1%

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

   4. frac-times 93.5%

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

   5. *-un-lft-identity 93.5%

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

   6. +-commutative 93.5%

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

   7. *-commutative 93.5%

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

   8. associate-+r+ 93.5%

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

6. Applied egg-rr 93.5%

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

   1. associate-/r* 97.1%

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

   2. associate-/l* 94.2%

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

   3. associate-*l/ 97.1%

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

   4. *-commutative 97.1%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.1%

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

   7. *-commutative 97.1%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in alpha around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 4: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 49000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{t_0 \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t_0} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 49000000.0)
     (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* t_0 (+ beta 3.0)))
     (* (/ (+ 1.0 alpha) t_0) (/ 1.0 (+ (+ beta 4.0) (* alpha 2.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 49000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 49000000.0d0) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / (t_0 * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / t_0) * (1.0d0 / ((beta + 4.0d0) + (alpha * 2.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 49000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 49000000.0:
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 49000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(t_0 * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 49000000.0)
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 49000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9e7

   1. Initial program 99.9%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in alpha around 0 86.0%

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

   6. Taylor expanded in alpha around 0 68.3%

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

   if 4.9e7 < beta

   1. Initial program 81.8%

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

   3. Simplified 90.2%

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

      1. clear-num 90.1%

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

      2. inv-pow 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. unpow-1 90.1%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 86.3%

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

       1. associate-+r+ 86.3%

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

       2. *-commutative 86.3%

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

   11. Simplified 86.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 49000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.52 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.52e+16)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ alpha (+ beta 2.0)) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.52d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((alpha + (beta + 2.0d0)) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.52e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.52e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.52e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((alpha + (beta + 2.0)) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.52e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.52e16

   1. Initial program 99.9%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in alpha around 0 85.2%

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

   6. Taylor expanded in alpha around 0 67.7%

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

   if 1.52e16 < beta

   1. Initial program 81.1%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around inf 86.5%

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

      1. un-div-inv 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-+r+ 86.6%

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

      4. +-commutative 86.6%

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

      5. +-commutative 86.6%

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

   7. Applied egg-rr 86.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.52 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $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. clear-num 97.1%

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

   2. associate-+r+ 97.1%

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

   3. *-commutative 97.1%

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

   4. frac-times 93.5%

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

   5. *-un-lft-identity 93.5%

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

   6. +-commutative 93.5%

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

   7. *-commutative 93.5%

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

   8. associate-+r+ 93.5%

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

6. Applied egg-rr 93.5%

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

   1. associate-/r* 97.1%

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

   2. associate-/l* 94.2%

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

   3. associate-*l/ 97.1%

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

   4. *-commutative 97.1%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.1%

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

   7. *-commutative 97.1%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in alpha around 0 99.8%

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

10. Taylor expanded in alpha around 0 70.9%

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

    1. associate-/r* 72.1%

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

12. Simplified 72.1%

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

13. Final simplification 72.1%

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

Alternative 7: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (- alpha -1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.0d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 97.2%

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

       1. +-commutative 97.2%

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

       2. +-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in alpha around 0 64.5%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   13. Step-by-step derivation

       1. *-commutative 64.5%

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

   14. Simplified 64.5%

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

   if 5 < beta

   1. Initial program 82.1%

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

   3. Taylor expanded in beta around -inf 83.2%

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

      1. expm1-log1p-u 83.2%

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

      2. expm1-udef 50.2%

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

      3. mul-1-neg 50.2%

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

      4. *-commutative 50.2%

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

      5. fma-neg 50.2%

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

      6. metadata-eval 50.2%

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

      7. metadata-eval 50.2%

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

      8. associate-+l+ 50.2%

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

      9. metadata-eval 50.2%

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

      10. associate-+r+ 50.2%

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

      11. +-commutative 50.2%

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

      12. associate-+r+ 50.2%

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

   5. Applied egg-rr 50.2%

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

      1. expm1-def 83.2%

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

      2. expm1-log1p 83.2%

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

      3. distribute-neg-frac 83.2%

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

      4. fma-udef 83.2%

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

      5. distribute-lft1-in 83.2%

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

      6. +-commutative 83.2%

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

      7. *-commutative 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. metadata-eval 83.2%

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

      10. neg-mul-1 83.2%

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

      11. unsub-neg 83.2%

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

      12. associate-+l+ 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.85e+16)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.85e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.85d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.85e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.85e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.85e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.85e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.85e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.85e16

   1. Initial program 99.9%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in alpha around 0 85.2%

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

   6. Taylor expanded in alpha around 0 66.8%

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

   if 1.85e16 < beta

   1. Initial program 81.1%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around inf 86.5%

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

      1. un-div-inv 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-+r+ 86.6%

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

      4. +-commutative 86.6%

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

      5. +-commutative 86.6%

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

   7. Applied egg-rr 86.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / 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 = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	}
	return tmp;
}

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 97.2%

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

       1. +-commutative 97.2%

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

       2. +-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in alpha around 0 64.8%

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

   if 6 < beta

   1. Initial program 82.1%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in beta around inf 83.0%

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

      1. un-div-inv 83.2%

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

      2. +-commutative 83.2%

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

      3. associate-+r+ 83.2%

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

      4. +-commutative 83.2%

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

      5. +-commutative 83.2%

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 70.0%

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

Alternative 10: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ (/ (- alpha -1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.0d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((alpha - -1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.0], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 97.2%

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

       1. +-commutative 97.2%

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

       2. +-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in alpha around 0 64.8%

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

   if 5 < beta

   1. Initial program 82.1%

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

   3. Taylor expanded in beta around -inf 83.2%

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

      1. expm1-log1p-u 83.2%

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

      2. expm1-udef 50.2%

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

      3. mul-1-neg 50.2%

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

      4. *-commutative 50.2%

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

      5. fma-neg 50.2%

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

      6. metadata-eval 50.2%

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

      7. metadata-eval 50.2%

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

      8. associate-+l+ 50.2%

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

      9. metadata-eval 50.2%

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

      10. associate-+r+ 50.2%

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

      11. +-commutative 50.2%

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

      12. associate-+r+ 50.2%

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

   5. Applied egg-rr 50.2%

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

      1. expm1-def 83.2%

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

      2. expm1-log1p 83.2%

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

      3. distribute-neg-frac 83.2%

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

      4. fma-udef 83.2%

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

      5. distribute-lft1-in 83.2%

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

      6. +-commutative 83.2%

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

      7. *-commutative 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. metadata-eval 83.2%

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

      10. neg-mul-1 83.2%

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

      11. unsub-neg 83.2%

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

      12. associate-+l+ 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 70.0%

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

Alternative 11: 96.5% 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 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 97.2%

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

       1. +-commutative 97.2%

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

       2. +-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in alpha around 0 64.8%

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

   if 7.5999999999999996 < beta

   1. Initial program 82.1%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in beta around inf 83.0%

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

   6. Taylor expanded in beta around inf 82.8%

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

4. Final simplification 69.9%

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

Alternative 12: 96.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ (/ (- alpha -1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((alpha - -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 <= 5.2d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((alpha - (-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 <= 5.2) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = ((alpha - -1.0) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / 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 <= 5.2)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((alpha - -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, 5.2], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 97.2%

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

       1. +-commutative 97.2%

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

       2. +-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in alpha around 0 64.8%

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

   if 5.20000000000000018 < beta

   1. Initial program 82.1%

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

   3. Taylor expanded in beta around -inf 83.2%

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

   4. Taylor expanded in alpha around 0 83.0%

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

      1. expm1-log1p-u 83.0%

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

      2. expm1-udef 46.0%

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

      3. mul-1-neg 46.0%

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

      4. *-commutative 46.0%

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

      5. fma-neg 46.0%

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

      6. metadata-eval 46.0%

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

      7. +-commutative 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. expm1-def 83.0%

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

      2. expm1-log1p 83.0%

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

      3. +-commutative 83.0%

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

      4. distribute-neg-frac 83.0%

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

      5. fma-udef 83.0%

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

      6. *-commutative 83.0%

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

      7. mul-1-neg 83.0%

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

      8. metadata-eval 83.0%

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

      9. distribute-neg-in 83.0%

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

      10. +-commutative 83.0%

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

      11. mul-1-neg 83.0%

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

      12. distribute-lft-in 83.0%

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

      13. metadata-eval 83.0%

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

      14. mul-1-neg 83.0%

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

      15. unsub-neg 83.0%

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

   8. Simplified 83.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ 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 <= 5.2) {
		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 <= 5.2d0) 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 <= 5.2) {
		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 <= 5.2:
		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 <= 5.2)
		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 <= 5.2)
		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, 5.2], 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 < 5.20000000000000018

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 97.2%

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

       1. +-commutative 97.2%

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

       2. +-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in alpha around 0 64.8%

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

   if 5.20000000000000018 < beta

   1. Initial program 82.1%

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

   3. Taylor expanded in beta around -inf 83.2%

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

   4. Taylor expanded in alpha around 0 72.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3e+122)
   (/ 0.16666666666666666 (+ beta 2.0))
   (* 0.3333333333333333 (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+122) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 0.3333333333333333 * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.3d+122) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = 0.3333333333333333d0 * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+122) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 0.3333333333333333 * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.3e+122:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = 0.3333333333333333 * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3e+122)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3e+122)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = 0.3333333333333333 * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.3e+122], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.30000000000000004e122

   1. Initial program 98.9%

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

   3. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow-1 99.3%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 86.4%

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

       1. +-commutative 86.4%

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

       2. +-commutative 86.4%

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

   11. Simplified 86.4%

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

   12. Taylor expanded in alpha around 0 57.0%

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

   if 1.30000000000000004e122 < beta

   1. Initial program 74.8%

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

   3. Taylor expanded in beta around -inf 93.4%

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

   4. Taylor expanded in alpha around 0 93.3%

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

   5. Taylor expanded in beta around 0 6.2%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-1 \cdot \alpha - 1}{\beta}} \]
   6. Step-by-step derivation

      1. sub-neg 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot \alpha + \left(-1\right)}}{\beta} \]

      2. mul-1-neg 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\left(-\alpha\right)} + \left(-1\right)}{\beta} \]

      3. distribute-neg-in 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-\left(\alpha + 1\right)}}{\beta} \]

      4. +-commutative 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{-\color{blue}{\left(1 + \alpha\right)}}{\beta} \]

      5. mul-1-neg 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot \left(1 + \alpha\right)}}{\beta} \]

      6. distribute-lft-in 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \alpha}}{\beta} \]

      7. metadata-eval 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1} + -1 \cdot \alpha}{\beta} \]

      8. mul-1-neg 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{-1 + \color{blue}{\left(-\alpha\right)}}{\beta} \]

      9. unsub-neg 6.2%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 - \alpha}}{\beta} \]

   7. Simplified 6.2%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-1 - \alpha}{\beta}} \]

   8. Taylor expanded in alpha around inf 44.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\alpha}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 19.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.3333333333333333 \cdot \frac{\alpha}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (* 0.3333333333333333 (/ alpha beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 * (alpha / beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 * (alpha / beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 * (alpha / beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 * (alpha / beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 * Float64(alpha / beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 * (alpha / beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in beta around -inf 25.7%

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

4. Taylor expanded in alpha around 0 25.5%

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

5. Taylor expanded in beta around 0 4.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-1 \cdot \alpha - 1}{\beta}} \]
6. Step-by-step derivation

   1. sub-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot \alpha + \left(-1\right)}}{\beta} \]

   2. mul-1-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\left(-\alpha\right)} + \left(-1\right)}{\beta} \]

   3. distribute-neg-in 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-\left(\alpha + 1\right)}}{\beta} \]

   4. +-commutative 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{-\color{blue}{\left(1 + \alpha\right)}}{\beta} \]

   5. mul-1-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot \left(1 + \alpha\right)}}{\beta} \]

   6. distribute-lft-in 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \alpha}}{\beta} \]

   7. metadata-eval 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1} + -1 \cdot \alpha}{\beta} \]

   8. mul-1-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{-1 + \color{blue}{\left(-\alpha\right)}}{\beta} \]

   9. unsub-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 - \alpha}}{\beta} \]

7. Simplified 4.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-1 - \alpha}{\beta}} \]

8. Taylor expanded in alpha around inf 10.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\alpha}{\beta}} \]

9. Final simplification 10.4%

   \[\leadsto 0.3333333333333333 \cdot \frac{\alpha}{\beta} \]
10. Add Preprocessing

Alternative 16: 6.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in beta around -inf 25.7%

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

4. Taylor expanded in alpha around 0 25.5%

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

5. Taylor expanded in beta around 0 4.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-1 \cdot \alpha - 1}{\beta}} \]
6. Step-by-step derivation

   1. sub-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot \alpha + \left(-1\right)}}{\beta} \]

   2. mul-1-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\left(-\alpha\right)} + \left(-1\right)}{\beta} \]

   3. distribute-neg-in 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-\left(\alpha + 1\right)}}{\beta} \]

   4. +-commutative 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{-\color{blue}{\left(1 + \alpha\right)}}{\beta} \]

   5. mul-1-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot \left(1 + \alpha\right)}}{\beta} \]

   6. distribute-lft-in 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \alpha}}{\beta} \]

   7. metadata-eval 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1} + -1 \cdot \alpha}{\beta} \]

   8. mul-1-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{-1 + \color{blue}{\left(-\alpha\right)}}{\beta} \]

   9. unsub-neg 4.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{-1 - \alpha}}{\beta} \]

7. Simplified 4.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-1 - \alpha}{\beta}} \]

8. Taylor expanded in alpha around 0 4.5%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]

9. Final simplification 4.5%

   \[\leadsto \frac{0.3333333333333333}{\beta} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024023 
(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)))