Octave 3.8, jcobi/3

Percentage Accurate: 94.7% → 99.8%

Time: 14.9s

Alternatives: 17

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: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 82.8%

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

   1. times-frac 95.2%

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

   2. +-commutative 95.2%

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

6. Applied egg-rr 95.2%

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

   1. associate-*r/ 95.3%

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

   2. +-commutative 95.3%

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

   3. +-commutative 95.3%

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

   4. +-commutative 95.3%

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

   5. +-commutative 95.3%

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

   6. +-commutative 95.3%

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

   7. +-commutative 95.3%

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

   8. +-commutative 95.3%

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

   9. associate-+r+ 95.3%

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

   10. +-commutative 95.3%

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

   11. +-commutative 95.3%

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

8. Simplified 95.3%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1 - \alpha}{t\_1} \cdot \frac{-1 - \beta}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{t\_0} \cdot \frac{\frac{\alpha}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

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 (+ alpha (+ beta 2.0))))
   (if (<= alpha 2.55e+14)
     (* (/ (- -1.0 alpha) t_1) (/ (- -1.0 beta) (* t_1 t_0)))
     (* (/ (+ 1.0 beta) t_0) (/ (/ alpha t_1) t_1)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (alpha <= 2.55e+14) {
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + beta) / t_0) * ((alpha / t_1) / 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) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (alpha <= 2.55d+14) then
        tmp = (((-1.0d0) - alpha) / t_1) * (((-1.0d0) - beta) / (t_1 * t_0))
    else
        tmp = ((1.0d0 + beta) / t_0) * ((alpha / t_1) / 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 = alpha + (beta + 2.0);
	double tmp;
	if (alpha <= 2.55e+14) {
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + beta) / t_0) * ((alpha / t_1) / t_1);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if alpha <= 2.55e+14:
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0))
	else:
		tmp = ((1.0 + beta) / t_0) * ((alpha / t_1) / 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(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (alpha <= 2.55e+14)
		tmp = Float64(Float64(Float64(-1.0 - alpha) / t_1) * Float64(Float64(-1.0 - beta) / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(Float64(alpha / t_1) / 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 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (alpha <= 2.55e+14)
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	else
		tmp = ((1.0 + beta) / t_0) * ((alpha / t_1) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.55e+14], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(-1.0 - beta), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(alpha / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.55e14

   1. Initial program 99.8%

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

   3. Simplified 93.8%

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

      1. times-frac 99.4%

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

      2. +-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   if 2.55e14 < alpha

   1. Initial program 82.4%

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

   3. Simplified 64.5%

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

      1. times-frac 88.3%

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

      2. +-commutative 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. associate-*r/ 88.3%

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

      2. +-commutative 88.3%

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

      3. +-commutative 88.3%

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

      4. +-commutative 88.3%

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

      5. +-commutative 88.3%

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

      6. +-commutative 88.3%

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

      7. +-commutative 88.3%

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

      8. +-commutative 88.3%

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

      9. associate-+r+ 88.3%

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

      10. +-commutative 88.3%

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

      11. +-commutative 88.3%

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

   8. Simplified 88.3%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in alpha around inf 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1 - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{-1 - \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{\alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{t\_0}}{t\_0 \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.75e+15)
     (/ (/ (+ 1.0 (+ alpha beta)) t_0) (* t_0 (+ 3.0 (+ alpha beta))))
     (/ (/ (+ 1.0 alpha) t_0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.75e+15) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.75d+15) then
        tmp = ((1.0d0 + (alpha + beta)) / t_0) / (t_0 * (3.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_0
    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 <= 1.75e+15) {
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.75e+15:
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / t_0) / t_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 <= 1.75e+15)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + beta)) / t_0) / Float64(t_0 * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_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 <= 1.75e+15)
		tmp = ((1.0 + (alpha + beta)) / t_0) / (t_0 * (3.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / t_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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.75e+15], N[(N[(N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75e15

   1. Initial program 99.8%

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

      1. associate-/l/ 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. associate-+l+ 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. associate-+l+ 99.7%

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

   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 98.6%

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

   if 1.75e15 < beta

   1. Initial program 80.3%

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

   3. Simplified 61.4%

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

      1. times-frac 86.5%

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

      2. +-commutative 86.5%

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

   6. Applied egg-rr 86.5%

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

      1. associate-*r/ 86.7%

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

      2. +-commutative 86.7%

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

      3. +-commutative 86.7%

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

      4. +-commutative 86.7%

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

      5. +-commutative 86.7%

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

      6. +-commutative 86.7%

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

      7. +-commutative 86.7%

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

      8. +-commutative 86.7%

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

      9. associate-+r+ 86.7%

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

      10. +-commutative 86.7%

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

      11. +-commutative 86.7%

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

   8. Simplified 86.7%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-commutative 99.7%

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

       2. clear-num 99.6%

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

       3. frac-times 98.9%

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

       4. *-un-lft-identity 98.9%

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

       5. associate-+r+ 98.9%

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

       6. +-commutative 98.9%

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

       7. associate-+l+ 98.9%

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

   12. Applied egg-rr 98.9%

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

   13. Taylor expanded in beta around inf 80.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.75e+15)
     (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
     (/ (/ (+ 1.0 alpha) t_0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.75e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.75d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_0
    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 <= 1.75e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.75e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / t_0) / t_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 <= 1.75e+15)
		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) / t_0) / t_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 <= 1.75e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / t_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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.75e+15], 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] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75e15

   1. Initial program 99.8%

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

   3. Simplified 93.7%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in alpha around 0 58.5%

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

       1. +-commutative 58.5%

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

       2. *-commutative 58.5%

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

       3. +-commutative 58.5%

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

   14. Simplified 58.5%

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

   if 1.75e15 < beta

   1. Initial program 80.3%

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

   3. Simplified 61.4%

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

      1. times-frac 86.5%

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

      2. +-commutative 86.5%

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

   6. Applied egg-rr 86.5%

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

      1. associate-*r/ 86.7%

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

      2. +-commutative 86.7%

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

      3. +-commutative 86.7%

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

      4. +-commutative 86.7%

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

      5. +-commutative 86.7%

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

      6. +-commutative 86.7%

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

      7. +-commutative 86.7%

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

      8. +-commutative 86.7%

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

      9. associate-+r+ 86.7%

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

      10. +-commutative 86.7%

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

      11. +-commutative 86.7%

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

   8. Simplified 86.7%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-commutative 99.7%

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

       2. clear-num 99.6%

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

       3. frac-times 98.9%

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

       4. *-un-lft-identity 98.9%

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

       5. associate-+r+ 98.9%

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

       6. +-commutative 98.9%

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

       7. associate-+l+ 98.9%

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

   12. Applied egg-rr 98.9%

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

   13. Taylor expanded in beta around inf 80.5%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.7× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+16)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (* beta (- (/ (- alpha -3.0) beta) -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	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 <= 1d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta * (((alpha - (-3.0d0)) / beta) - (-1.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+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) / beta) / Float64(beta * Float64(Float64(Float64(alpha - -3.0) / beta) - -1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	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, 1e+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] / beta), $MachinePrecision] / N[(beta * N[(N[(N[(alpha - -3.0), $MachinePrecision] / beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e16

   1. Initial program 99.8%

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

   3. Simplified 93.7%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in alpha around 0 58.5%

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

       1. +-commutative 58.5%

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

       2. *-commutative 58.5%

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

       3. +-commutative 58.5%

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

   14. Simplified 58.5%

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

   if 1e16 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 80.0%

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

   4. Taylor expanded in beta around -inf 80.0%

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

      1. associate-*r* 80.0%

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

      2. mul-1-neg 80.0%

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

      3. sub-neg 80.0%

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

      4. associate-*r/ 80.0%

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

      5. distribute-lft-in 80.0%

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

      6. metadata-eval 80.0%

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

      7. metadata-eval 80.0%

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

      8. mul-1-neg 80.0%

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

      9. unsub-neg 80.0%

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

      10. metadata-eval 80.0%

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

      11. metadata-eval 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 65.7%

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

Alternative 6: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.4d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4e+15)
		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) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4e+15], 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] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4e15

   1. Initial program 99.8%

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

   3. Simplified 93.7%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in alpha around 0 58.5%

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

       1. +-commutative 58.5%

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

       2. *-commutative 58.5%

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

       3. +-commutative 58.5%

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

   14. Simplified 58.5%

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

   if 5.4e15 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 80.0%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. associate-+l+ 79.9%

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

      4. metadata-eval 79.9%

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

      5. associate-+r+ 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. associate-*r/ 80.0%

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

      2. *-rgt-identity 80.0%

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

      3. associate-+r+ 80.0%

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

      4. +-commutative 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 21:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 21.0)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 21.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 21.0d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 21.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 21.0:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 21.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 21.0)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 21.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 21

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around 0 96.7%

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

   10. Taylor expanded in beta around 0 96.8%

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

   if 21 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

      1. div-inv 76.0%

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

      2. metadata-eval 76.0%

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

      3. associate-+l+ 76.0%

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

      4. metadata-eval 76.0%

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

      5. associate-+r+ 76.0%

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

   5. Applied egg-rr 76.0%

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

      1. associate-*r/ 76.1%

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

      2. *-rgt-identity 76.1%

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

      3. associate-+r+ 76.1%

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

      4. +-commutative 76.1%

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

   7. Simplified 76.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 21:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ alpha beta))))
   (if (<= beta 4.8)
     (/ 0.5 (* (+ beta 2.0) t_0))
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.5 / ((beta + 2.0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (alpha + beta)
    if (beta <= 4.8d0) then
        tmp = 0.5d0 / ((beta + 2.0d0) * t_0)
    else
        tmp = ((1.0d0 + alpha) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.5 / ((beta + 2.0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (alpha + beta)
	tmp = 0
	if beta <= 4.8:
		tmp = 0.5 / ((beta + 2.0) * t_0)
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(0.5 / Float64(Float64(beta + 2.0) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 4.8)
		tmp = 0.5 / ((beta + 2.0) * t_0);
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.8], N[(0.5 / N[(N[(beta + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in beta around 0 80.5%

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

   13. Taylor expanded in alpha around 0 58.3%

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

       1. +-commutative 58.3%

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

   15. Simplified 58.3%

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

   if 4.79999999999999982 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

      1. div-inv 76.0%

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

      2. metadata-eval 76.0%

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

      3. associate-+l+ 76.0%

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

      4. metadata-eval 76.0%

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

      5. associate-+r+ 76.0%

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

   5. Applied egg-rr 76.0%

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

      1. associate-*r/ 76.1%

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

      2. *-rgt-identity 76.1%

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

      3. associate-+r+ 76.1%

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

      4. +-commutative 76.1%

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

   7. Simplified 76.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5)
   (/ 0.5 (* (+ beta 2.0) (+ 3.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.5 / ((beta + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d0) then
        tmp = 0.5d0 / ((beta + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.5 / ((beta + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5:
		tmp = 0.5 / ((beta + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5)
		tmp = Float64(0.5 / Float64(Float64(beta + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5)
		tmp = 0.5 / ((beta + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5], N[(0.5 / N[(N[(beta + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in beta around 0 80.5%

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

   13. Taylor expanded in alpha around 0 58.3%

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

       1. +-commutative 58.3%

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

   15. Simplified 58.3%

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

   if 6.5 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

   4. Taylor expanded in alpha around 0 75.8%

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

      1. +-commutative 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 24.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 24.5)
   (/ 0.5 (* (+ alpha 2.0) (+ 3.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 24.5) {
		tmp = 0.5 / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 24.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 24.5) {
		tmp = 0.5 / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 24.5:
		tmp = 0.5 / ((alpha + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 24.5)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 24.5)
		tmp = 0.5 / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 24.5], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 24.5

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in beta around 0 80.5%

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

   13. Taylor expanded in beta around 0 81.4%

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

   if 24.5 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

   4. Taylor expanded in alpha around 0 75.8%

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

      1. +-commutative 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 24.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6)
   (/ 0.5 (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = 0.5d0 / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(0.5 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(0.5 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5999999999999996

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in beta around 0 80.5%

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

   13. Taylor expanded in alpha around 0 56.4%

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

       1. +-commutative 58.7%

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

       2. *-commutative 58.7%

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

       3. +-commutative 58.7%

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

   15. Simplified 56.4%

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

   if 4.5999999999999996 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

   4. Taylor expanded in alpha around 0 75.8%

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

      1. +-commutative 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.8:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.8)
   (/ 0.5 (* (+ beta 2.0) (+ beta 3.0)))
   (/ 1.0 (* beta (/ beta (+ 1.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.8) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.8d0) then
        tmp = 0.5d0 / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = 1.0d0 / (beta * (beta / (1.0d0 + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.8) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.8:
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.8)
		tmp = Float64(0.5 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta / Float64(1.0 + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.8)
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = 1.0 / (beta * (beta / (1.0 + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.8], N[(0.5 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.8000000000000007

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in beta around 0 80.5%

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

   13. Taylor expanded in alpha around 0 56.4%

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

       1. +-commutative 58.7%

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

       2. *-commutative 58.7%

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

       3. +-commutative 58.7%

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

   15. Simplified 56.4%

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

   if 9.8000000000000007 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

      1. clear-num 75.6%

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

      2. inv-pow 75.6%

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

      3. metadata-eval 75.6%

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

      4. associate-+l+ 75.6%

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

      5. metadata-eval 75.6%

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

      6. associate-+r+ 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. unpow-1 75.6%

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

      2. associate-/r/ 75.7%

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

      3. associate-+r+ 75.7%

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

      4. +-commutative 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in beta around inf 75.3%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.8:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ 0.5 (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} 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 <= 4.5d0) then
        tmp = 0.5d0 / ((beta + 2.0d0) * (beta + 3.0d0))
    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 <= 4.5) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0))
	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 <= 4.5)
		tmp = Float64(0.5 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(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 <= 4.5)
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	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, 4.5], N[(0.5 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in beta around 0 80.5%

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

   13. Taylor expanded in alpha around 0 56.4%

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

       1. +-commutative 58.7%

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

       2. *-commutative 58.7%

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

       3. +-commutative 58.7%

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

   15. Simplified 56.4%

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

   if 4.5 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

      1. clear-num 75.6%

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

      2. inv-pow 75.6%

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

      3. metadata-eval 75.6%

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

      4. associate-+l+ 75.6%

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

      5. metadata-eval 75.6%

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

      6. associate-+r+ 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. unpow-1 75.6%

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

      2. associate-/r/ 75.7%

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

      3. associate-+r+ 75.7%

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

      4. +-commutative 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in alpha around 0 70.9%

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

      1. +-commutative 70.9%

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

   10. Simplified 70.9%

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

       1. *-un-lft-identity 70.9%

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

       2. *-commutative 70.9%

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

   12. Applied egg-rr 70.9%

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

       1. *-lft-identity 70.9%

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

       2. associate-/r* 71.4%

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

   14. Simplified 71.4%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 33:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 33.0)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 33.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} 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 <= 33.0d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    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 <= 33.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 33.0:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	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 <= 33.0)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(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 <= 33.0)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	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, 33.0], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 33

   1. Initial program 99.8%

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

   3. Simplified 94.0%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   12. Taylor expanded in beta around 0 80.6%

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

       1. +-commutative 80.6%

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

   14. Simplified 80.6%

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

   if 33 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf 76.1%

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

      1. clear-num 75.6%

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

      2. inv-pow 75.6%

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

      3. metadata-eval 75.6%

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

      4. associate-+l+ 75.6%

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

      5. metadata-eval 75.6%

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

      6. associate-+r+ 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. unpow-1 75.6%

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

      2. associate-/r/ 75.7%

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

      3. associate-+r+ 75.7%

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

      4. +-commutative 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in alpha around 0 70.9%

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

      1. +-commutative 70.9%

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

   10. Simplified 70.9%

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

       1. *-un-lft-identity 70.9%

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

       2. *-commutative 70.9%

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

   12. Applied egg-rr 70.9%

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

       1. *-lft-identity 70.9%

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

       2. associate-/r* 71.4%

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

   14. Simplified 71.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 33:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in beta around inf 29.3%

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

   1. clear-num 29.1%

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

   2. inv-pow 29.1%

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

   3. metadata-eval 29.1%

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

   4. associate-+l+ 29.1%

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

   5. metadata-eval 29.1%

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

   6. associate-+r+ 29.1%

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

5. Applied egg-rr 29.1%

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

   1. unpow-1 29.1%

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

   2. associate-/r/ 29.2%

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

   3. associate-+r+ 29.2%

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

   4. +-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in alpha around 0 27.5%

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

   1. +-commutative 27.5%

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

10. Simplified 27.5%

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

    1. *-un-lft-identity 27.5%

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

    2. *-commutative 27.5%

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

12. Applied egg-rr 27.5%

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

    1. *-lft-identity 27.5%

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

    2. associate-/r* 27.7%

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

14. Simplified 27.7%

    \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{\beta + 3}} \]
15. Add Preprocessing

Alternative 16: 49.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in beta around inf 29.3%

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

4. Taylor expanded in alpha around 0 27.5%

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

   1. +-commutative 27.5%

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

6. Simplified 27.5%

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

Alternative 17: 6.0% 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 93.3%

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

3. Taylor expanded in beta around inf 29.3%

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

   1. clear-num 29.1%

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

   2. inv-pow 29.1%

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

   3. metadata-eval 29.1%

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

   4. associate-+l+ 29.1%

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

   5. metadata-eval 29.1%

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

   6. associate-+r+ 29.1%

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

5. Applied egg-rr 29.1%

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

   1. unpow-1 29.1%

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

   2. associate-/r/ 29.2%

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

   3. associate-+r+ 29.2%

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

   4. +-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in alpha around 0 27.5%

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

   1. +-commutative 27.5%

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

10. Simplified 27.5%

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

11. Taylor expanded in beta around 0 4.5%

    \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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)))