Octave 3.8, jcobi/3

Percentage Accurate: 93.9% → 99.8%

Time: 13.6s

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: 93.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $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 94.8%

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

3. Simplified 85.1%

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

   1. times-frac 97.1%

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

   2. +-commutative 97.1%

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

6. Applied egg-rr 97.1%

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

   1. associate-*r/ 97.1%

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

8. Applied egg-rr 97.1%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1020000000:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 1020000000.0)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* (+ alpha (+ beta 3.0)) t_1)))
     (* (/ (/ (+ 1.0 alpha) t_0) t_0) (+ 1.0 (/ (- -2.0 alpha) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1020000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_1));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 1020000000.0d0) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / ((alpha + (beta + 3.0d0)) * t_1))
    else
        tmp = (((1.0d0 + alpha) / t_0) / t_0) * (1.0d0 + (((-2.0d0) - alpha) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1020000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_1));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 1020000000.0:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_1))
	else:
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 1020000000.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0) * Float64(1.0 + Float64(Float64(-2.0 - alpha) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 1020000000.0)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_1));
	else
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1020000000.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.02e9

   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 94.5%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 1.02e9 < beta

   1. Initial program 83.2%

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

   3. Simplified 63.1%

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

      1. times-frac 91.3%

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

      2. +-commutative 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. associate-*r/ 91.3%

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

   8. Applied egg-rr 91.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in beta around inf 84.9%

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

       1. associate-*r/ 84.9%

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

       2. distribute-lft-in 84.9%

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

       3. metadata-eval 84.9%

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

       4. neg-mul-1 84.9%

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

   13. Simplified 84.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1020000000:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 15.5:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 15.5)
     (/
      (/ (+ 1.0 alpha) (+ alpha 2.0))
      (* (+ alpha 2.0) (+ 3.0 (+ alpha beta))))
     (* (/ (/ (+ 1.0 alpha) t_0) t_0) (+ 1.0 (/ (- -2.0 alpha) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 15.5) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 15.5d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = (((1.0d0 + alpha) / t_0) / t_0) * (1.0d0 + (((-2.0d0) - alpha) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 15.5) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 15.5:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 15.5)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0) * Float64(1.0 + Float64(Float64(-2.0 - alpha) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 15.5)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 15.5], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 15.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in beta around 0 99.0%

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

   if 15.5 < beta

   1. Initial program 83.8%

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

   3. Simplified 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 91.6%

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

      2. +-commutative 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. associate-*r/ 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in beta around inf 83.6%

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

       1. associate-*r/ 83.6%

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

       2. distribute-lft-in 83.6%

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

       3. metadata-eval 83.6%

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

       4. neg-mul-1 83.6%

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

   13. Simplified 83.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 15.5:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.8)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ 3.0 (+ alpha beta))))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.8d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.8:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.8)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.79999999999999982

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in beta around 0 99.0%

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

   if 6.79999999999999982 < beta

   1. Initial program 83.8%

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

   3. Simplified 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 91.6%

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

      2. +-commutative 91.6%

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

   6. Applied egg-rr 91.6%

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

   7. Taylor expanded in beta around inf 83.2%

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

      1. mul-1-neg 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \frac{1 + \beta}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.8e+15)
     (/ (+ 1.0 beta) (* (+ alpha (+ 2.0 beta)) (* (+ beta 3.0) (+ 2.0 beta))))
     (* (/ (/ (+ 1.0 alpha) t_0) t_0) (/ (+ 1.0 beta) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.8e+15) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 3.8d+15) then
        tmp = (1.0d0 + beta) / ((alpha + (2.0d0 + beta)) * ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = (((1.0d0 + alpha) / t_0) / t_0) * ((1.0d0 + beta) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.8e+15) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 3.8e+15:
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.8e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0) * Float64(Float64(1.0 + beta) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 3.8e+15)
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.8e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8e15

   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 94.5%

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

   5. Taylor expanded in alpha around 0 83.8%

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

   6. Taylor expanded in alpha around 0 71.5%

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

      1. +-commutative 71.5%

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

      2. +-commutative 71.5%

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

   8. Simplified 71.5%

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

   if 3.8e15 < beta

   1. Initial program 83.2%

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

   3. Simplified 63.1%

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

      1. times-frac 91.3%

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

      2. +-commutative 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. associate-*r/ 91.3%

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

   8. Applied egg-rr 91.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in beta around inf 85.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 + \beta}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.6× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3e+17)
   (/ (+ 1.0 beta) (* (+ alpha (+ 2.0 beta)) (* (+ beta 3.0) (+ 2.0 beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+17) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3d+17) then
        tmp = (1.0d0 + beta) / ((alpha + (2.0d0 + beta)) * ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+17) {
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3e+17:
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3e+17)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3e+17)
		tmp = (1.0 + beta) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3e+17], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e17

   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 94.5%

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

   5. Taylor expanded in alpha around 0 83.8%

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

   6. Taylor expanded in alpha around 0 71.5%

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

      1. +-commutative 71.5%

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

      2. +-commutative 71.5%

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

   8. Simplified 71.5%

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

   if 3e17 < beta

   1. Initial program 83.2%

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

   3. Taylor expanded in beta around inf 84.6%

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

      1. div-inv 84.5%

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

      2. +-commutative 84.5%

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

      3. metadata-eval 84.5%

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

      4. associate-+l+ 84.5%

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

      5. metadata-eval 84.5%

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

      6. associate-+r+ 84.5%

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

   5. Applied egg-rr 84.5%

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

      1. associate-*r/ 84.6%

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

      2. *-commutative 84.6%

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

      3. *-lft-identity 84.6%

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

      4. +-commutative 84.6%

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

      5. +-commutative 84.6%

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

      6. +-commutative 84.6%

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

      7. +-commutative 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 75.4%

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

Alternative 7: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 26:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 26.0)
   (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 26.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 26.0d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 26.0) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 26.0:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 26.0)
		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(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 26.0)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 26.0], 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[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 26

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in beta around 0 98.2%

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

   if 26 < beta

   1. Initial program 83.8%

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

   3. Taylor expanded in beta around inf 82.6%

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

      1. div-inv 82.5%

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

      2. +-commutative 82.5%

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

      3. metadata-eval 82.5%

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

      4. associate-+l+ 82.5%

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

      5. metadata-eval 82.5%

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

      6. associate-+r+ 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. associate-*r/ 82.6%

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

      2. *-commutative 82.6%

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

      3. *-lft-identity 82.6%

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

      4. +-commutative 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

      7. +-commutative 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 26:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot \beta + 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ 1.0 (* (+ 2.0 (+ alpha beta)) (+ (* 2.0 beta) 6.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * ((2.0 * beta) + 6.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 1.0d0 / ((2.0d0 + (alpha + beta)) * ((2.0d0 * beta) + 6.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 1.0 / ((2.0 + (alpha + beta)) * ((2.0 * beta) + 6.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 1.0 / ((2.0 + (alpha + beta)) * ((2.0 * beta) + 6.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(alpha + beta)) * Float64(Float64(2.0 * beta) + 6.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 1.0 / ((2.0 + (alpha + beta)) * ((2.0 * beta) + 6.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(1.0 / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * beta), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.2%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-+r+ 98.2%

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

      5. +-commutative 98.2%

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

      6. +-commutative 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. unpow-1 98.2%

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

      2. associate-/l* 98.2%

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

      3. associate-+r+ 98.2%

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

      4. +-commutative 98.2%

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

      5. +-commutative 98.2%

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

      6. +-commutative 98.2%

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

      7. +-commutative 98.2%

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

      8. +-commutative 98.2%

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

      9. +-commutative 98.2%

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

      10. +-commutative 98.2%

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

   9. Simplified 98.2%

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

   10. Taylor expanded in alpha around 0 70.1%

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

       1. distribute-lft-in 70.1%

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

       2. metadata-eval 70.1%

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

       3. +-commutative 70.1%

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

       4. *-commutative 70.1%

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

   12. Simplified 70.1%

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

   if 4.5 < beta

   1. Initial program 83.8%

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

   3. Taylor expanded in beta around inf 82.6%

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

      1. div-inv 82.5%

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

      2. +-commutative 82.5%

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

      3. metadata-eval 82.5%

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

      4. associate-+l+ 82.5%

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

      5. metadata-eval 82.5%

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

      6. associate-+r+ 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. associate-*r/ 82.6%

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

      2. *-commutative 82.6%

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

      3. *-lft-identity 82.6%

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

      4. +-commutative 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

      7. +-commutative 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot \beta + 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 0.5 (* (+ beta 3.0) (+ 2.0 beta)))
   (if (<= beta 4.8e+157)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.5 / ((beta + 3.0) * (2.0 + beta));
	} else if (beta <= 4.8e+157) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (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 <= 8.0d0) then
        tmp = 0.5d0 / ((beta + 3.0d0) * (2.0d0 + beta))
    else if (beta <= 4.8d+157) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (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 <= 8.0) {
		tmp = 0.5 / ((beta + 3.0) * (2.0 + beta));
	} else if (beta <= 4.8e+157) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 0.5 / ((beta + 3.0) * (2.0 + beta))
	elif beta <= 4.8e+157:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(0.5 / Float64(Float64(beta + 3.0) * Float64(2.0 + beta)));
	elseif (beta <= 4.8e+157)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.0)
		tmp = 0.5 / ((beta + 3.0) * (2.0 + beta));
	elseif (beta <= 4.8e+157)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (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, 8.0], N[(0.5 / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.8e+157], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in alpha around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. +-commutative 69.2%

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

   8. Simplified 69.2%

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

   if 8 < beta < 4.7999999999999999e157

   1. Initial program 88.7%

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

   3. Taylor expanded in beta around inf 75.3%

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

      1. *-un-lft-identity 75.3%

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

      2. associate-/l/ 83.7%

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

      3. +-commutative 83.7%

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

      4. metadata-eval 83.7%

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

      5. associate-+l+ 83.7%

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

      6. metadata-eval 83.7%

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

      7. associate-+r+ 83.7%

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

   5. Applied egg-rr 83.7%

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

      1. *-lft-identity 83.7%

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

      2. +-commutative 83.7%

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

      3. *-commutative 83.7%

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

      4. +-commutative 83.7%

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

      5. +-commutative 83.7%

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

      6. +-commutative 83.7%

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

   7. Simplified 83.7%

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

   8. Taylor expanded in beta around inf 74.9%

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

   if 4.7999999999999999e157 < beta

   1. Initial program 77.5%

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

   3. Taylor expanded in beta around inf 92.1%

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

   4. Taylor expanded in alpha around 0 92.0%

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

      1. +-commutative 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in alpha around inf 86.7%

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

      1. associate-/r* 90.6%

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

      2. +-commutative 90.6%

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

   9. Simplified 90.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.5d0 / ((beta + 3.0d0) * (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.5 / ((beta + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (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 + 3.0) * (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (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 + 3.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.5 / ((beta + 3.0) * (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(0.5 / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in alpha around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. +-commutative 69.2%

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

   8. Simplified 69.2%

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

   if 4.5 < beta

   1. Initial program 83.8%

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

   3. Taylor expanded in beta around inf 82.6%

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

      1. div-inv 82.5%

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

      2. +-commutative 82.5%

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

      3. metadata-eval 82.5%

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

      4. associate-+l+ 82.5%

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

      5. metadata-eval 82.5%

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

      6. associate-+r+ 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. associate-*r/ 82.6%

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

      2. *-commutative 82.6%

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

      3. *-lft-identity 82.6%

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

      4. +-commutative 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

      7. +-commutative 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 73.4%

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

Alternative 11: 96.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.5d0 / ((beta + 3.0d0) * (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(0.5 / Float64(Float64(beta + 3.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.5 / ((beta + 3.0) * (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(0.5 / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around 0 98.2%

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

   6. Taylor expanded in alpha around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. +-commutative 69.2%

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

   8. Simplified 69.2%

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

   if 4.5 < beta

   1. Initial program 83.8%

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

   3. Taylor expanded in beta around inf 82.6%

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

   4. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

   6. Simplified 82.5%

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

4. Final simplification 73.3%

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

Alternative 12: 54.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.2e-26)
   (/ (/ 1.0 beta) (+ beta 3.0))
   (/ (/ alpha beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.2e-26) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (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 (alpha <= 1.2d-26) then
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    else
        tmp = (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 (alpha <= 1.2e-26) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 1.2e-26:
		tmp = (1.0 / beta) / (beta + 3.0)
	else:
		tmp = (alpha / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.2e-26)
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.2e-26)
		tmp = (1.0 / beta) / (beta + 3.0);
	else
		tmp = (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[alpha, 1.2e-26], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.2e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf 31.7%

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

   4. Taylor expanded in alpha around 0 31.3%

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

      1. associate-/r* 31.7%

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

      2. +-commutative 31.7%

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

   6. Simplified 31.7%

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

   if 1.2e-26 < alpha

   1. Initial program 84.9%

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

   3. Taylor expanded in beta around inf 20.2%

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

   4. Taylor expanded in alpha around 0 19.8%

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

      1. +-commutative 19.8%

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

   6. Simplified 19.8%

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

   7. Taylor expanded in alpha around inf 18.4%

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

      1. associate-/r* 19.9%

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

      2. +-commutative 19.9%

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

   9. Simplified 19.9%

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

Alternative 13: 52.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta \cdot \left(\beta + 3\right)\\ \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (* beta (+ beta 3.0))))
   (if (<= alpha 1.2e-26) (/ 1.0 t_0) (/ alpha t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta * (beta + 3.0);
	double tmp;
	if (alpha <= 1.2e-26) {
		tmp = 1.0 / t_0;
	} else {
		tmp = alpha / 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 = beta * (beta + 3.0d0)
    if (alpha <= 1.2d-26) then
        tmp = 1.0d0 / t_0
    else
        tmp = alpha / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = beta * (beta + 3.0);
	double tmp;
	if (alpha <= 1.2e-26) {
		tmp = 1.0 / t_0;
	} else {
		tmp = alpha / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta * (beta + 3.0)
	tmp = 0
	if alpha <= 1.2e-26:
		tmp = 1.0 / t_0
	else:
		tmp = alpha / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(beta * Float64(beta + 3.0))
	tmp = 0.0
	if (alpha <= 1.2e-26)
		tmp = Float64(1.0 / t_0);
	else
		tmp = Float64(alpha / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = beta * (beta + 3.0);
	tmp = 0.0;
	if (alpha <= 1.2e-26)
		tmp = 1.0 / t_0;
	else
		tmp = alpha / 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[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 1.2e-26], N[(1.0 / t$95$0), $MachinePrecision], N[(alpha / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.2e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf 31.7%

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

   4. Taylor expanded in alpha around 0 31.3%

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

   if 1.2e-26 < alpha

   1. Initial program 84.9%

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

   3. Taylor expanded in beta around inf 20.2%

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

   4. Taylor expanded in alpha around 0 19.8%

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

      1. +-commutative 19.8%

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

   6. Simplified 19.8%

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

   7. Taylor expanded in alpha around inf 18.4%

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

      1. +-commutative 18.4%

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

   9. Simplified 18.4%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in beta around inf 27.8%

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

   1. *-un-lft-identity 27.8%

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

   2. associate-/l/ 28.6%

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

   3. +-commutative 28.6%

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

   4. metadata-eval 28.6%

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

   5. associate-+l+ 28.6%

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

   6. metadata-eval 28.6%

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

   7. associate-+r+ 28.6%

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

5. Applied egg-rr 28.6%

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

   1. *-lft-identity 28.6%

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

   2. +-commutative 28.6%

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

   3. *-commutative 28.6%

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

   4. +-commutative 28.6%

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

   5. +-commutative 28.6%

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

   6. +-commutative 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in beta around inf 27.5%

   \[\leadsto \frac{1 + \alpha}{\beta \cdot \color{blue}{\beta}} \]
9. Add Preprocessing

Alternative 15: 50.5% 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 94.8%

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

3. Taylor expanded in beta around inf 27.8%

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

4. Taylor expanded in alpha around 0 25.5%

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

5. Final simplification 25.5%

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

Alternative 16: 6.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in beta around inf 27.8%

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

   1. *-un-lft-identity 27.8%

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

   2. associate-/l/ 28.6%

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

   3. +-commutative 28.6%

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

   4. metadata-eval 28.6%

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

   5. associate-+l+ 28.6%

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

   6. metadata-eval 28.6%

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

   7. associate-+r+ 28.6%

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

5. Applied egg-rr 28.6%

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

   1. *-lft-identity 28.6%

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

   2. +-commutative 28.6%

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

   3. *-commutative 28.6%

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

   4. +-commutative 28.6%

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

   5. +-commutative 28.6%

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

   6. +-commutative 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in alpha around 0 25.5%

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

   1. +-commutative 25.5%

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

10. Simplified 25.5%

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

11. Taylor expanded in beta around 0 4.5%

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

    1. *-commutative 4.5%

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

13. Simplified 4.5%

    \[\leadsto \frac{1}{\color{blue}{\beta \cdot 3}} \]
14. 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 94.8%

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

3. Taylor expanded in beta around inf 27.8%

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

   1. *-un-lft-identity 27.8%

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

   2. associate-/l/ 28.6%

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

   3. +-commutative 28.6%

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

   4. metadata-eval 28.6%

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

   5. associate-+l+ 28.6%

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

   6. metadata-eval 28.6%

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

   7. associate-+r+ 28.6%

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

5. Applied egg-rr 28.6%

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

   1. *-lft-identity 28.6%

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

   2. +-commutative 28.6%

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

   3. *-commutative 28.6%

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

   4. +-commutative 28.6%

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

   5. +-commutative 28.6%

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

   6. +-commutative 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in alpha around 0 25.5%

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

   1. +-commutative 25.5%

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

10. Simplified 25.5%

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

11. Taylor expanded in beta around 0 4.2%

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

Reproduce

herbie shell --seed 2024170 
(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)))