Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 18.0s

Alternatives: 18

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 83.1%

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

   1. times-frac 95.5%

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

   2. +-commutative 95.5%

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

6. Applied egg-rr 95.5%

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

   1. associate-*l/ 95.5%

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

   2. +-commutative 95.5%

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

   3. associate-+r+ 95.5%

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

   4. associate-/r* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 160000000000:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 + \beta}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 160000000000.0)
     (* (/ (+ 1.0 alpha) t_0) (/ (+ 1.0 beta) (* t_0 (+ alpha (+ beta 3.0)))))
     (/
      (*
       (+ 1.0 alpha)
       (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ 3.0 (+ alpha beta))))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 160000000000.0) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 160000000000.0d0) then
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 + beta) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (3.0d0 + (alpha + beta)))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 160000000000.0) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 160000000000.0:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 160000000000.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(3.0 + Float64(alpha + beta)))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 160000000000.0)
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 160000000000.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6e11

   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 94.4%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 1.6e11 < beta

   1. Initial program 82.5%

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

   3. Simplified 60.4%

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

      1. times-frac 86.8%

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

      2. +-commutative 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. associate-*l/ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+r+ 86.7%

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

      4. associate-/r* 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in beta around inf 90.8%

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

       1. neg-mul-1 90.8%

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

       2. distribute-neg-frac2 90.8%

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

   11. Simplified 90.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 160000000000:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.2e+37)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/
      (*
       (+ 1.0 alpha)
       (/ (+ 1.0 (/ (- -1.0 alpha) beta)) (+ 3.0 (+ alpha beta))))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.2e+37) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3.2d+37) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 + (((-1.0d0) - alpha) / beta)) / (3.0d0 + (alpha + beta)))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.2e+37) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.2e+37:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.2e+37)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) / Float64(3.0 + Float64(alpha + beta)))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.2e+37)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) * ((1.0 + ((-1.0 - alpha) / beta)) / (3.0 + (alpha + beta)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.2e+37], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.20000000000000014e37

   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 94.6%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in alpha around 0 70.0%

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

       1. +-commutative 70.0%

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

       2. +-commutative 70.0%

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

   11. Simplified 70.0%

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

   if 3.20000000000000014e37 < beta

   1. Initial program 81.4%

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

   3. Simplified 58.0%

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

      1. times-frac 86.0%

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

      2. +-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*l/ 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-+r+ 85.9%

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

      4. associate-/r* 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in beta around inf 90.2%

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

       1. neg-mul-1 90.2%

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

       2. distribute-neg-frac2 90.2%

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

   11. Simplified 90.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 + \frac{-1 - \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+47)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (*
      (/ (+ 1.0 alpha) t_0)
      (/ (- 1.0 (/ (* 2.0 (+ alpha 2.0)) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+47) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((2.0 * (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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5d+47) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 - ((2.0d0 * (alpha + 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+47) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+47:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+47)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 - Float64(Float64(2.0 * Float64(alpha + 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+47)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((2.0 * (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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+47], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(2.0 * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000022e47

   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 94.6%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in alpha around 0 70.0%

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

       1. +-commutative 70.0%

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

       2. +-commutative 70.0%

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

   11. Simplified 70.0%

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

   if 5.00000000000000022e47 < beta

   1. Initial program 81.4%

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

   3. Simplified 58.0%

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

      1. times-frac 86.0%

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

      2. +-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in beta around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. metadata-eval 90.2%

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

      3. distribute-lft-in 90.2%

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

   9. Simplified 90.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+35)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/
      (* (+ 1.0 alpha) (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+35) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2d+35) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+35) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2e+35:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+35)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2e+35)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+35], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9999999999999999e35

   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 94.6%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in alpha around 0 70.0%

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

       1. +-commutative 70.0%

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

       2. +-commutative 70.0%

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

   11. Simplified 70.0%

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

   if 1.9999999999999999e35 < beta

   1. Initial program 81.4%

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

   3. Simplified 58.0%

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

      1. times-frac 86.0%

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

      2. +-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*l/ 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-+r+ 85.9%

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

      4. associate-/r* 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in beta around inf 90.2%

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

       1. mul-1-neg 90.2%

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

   11. Simplified 90.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.95e+18)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.95e+18) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.95d+18) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.95e+18) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.95e+18:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.95e+18)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.95e+18)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.95e+18], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.95e18

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+l+ 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.4%

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

      1. +-commutative 84.4%

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

   7. Simplified 84.4%

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

      1. add-exp-log 84.2%

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

      2. associate-/l/ 84.2%

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

      3. log-div 84.2%

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

      4. log1p-define 84.2%

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

      5. *-commutative 84.2%

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

      6. +-commutative 84.2%

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

      7. +-commutative 84.2%

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

   9. Applied egg-rr 84.2%

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

       1. exp-diff 84.2%

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

       2. log1p-undefine 84.2%

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

       3. rem-exp-log 84.2%

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

       4. rem-exp-log 84.3%

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

       5. associate-*l* 84.3%

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

       6. +-commutative 84.3%

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

       7. +-commutative 84.3%

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

       8. *-commutative 84.3%

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

       9. +-commutative 84.3%

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

   11. Simplified 84.3%

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

   12. Taylor expanded in alpha around 0 69.4%

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

       1. +-commutative 69.4%

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

       2. +-commutative 69.4%

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

   14. Simplified 69.4%

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

   if 1.95e18 < beta

   1. Initial program 82.1%

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

   3. Simplified 59.5%

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

      1. times-frac 86.5%

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

      2. +-commutative 86.5%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in beta around inf 90.1%

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

      1. frac-times 84.9%

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

      2. *-rgt-identity 84.9%

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

      3. +-commutative 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-/l/ 90.2%

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

       2. +-commutative 90.2%

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

       3. +-commutative 90.2%

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

       4. +-commutative 90.2%

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

   11. Simplified 90.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.95 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.95e+35)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.95e+35) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.95d+35) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.95e+35) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.95e+35:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.95e+35)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.95e+35)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.95e+35], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.94999999999999993e35

   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 94.6%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in alpha around 0 70.0%

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

       1. +-commutative 70.0%

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

       2. +-commutative 70.0%

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

   11. Simplified 70.0%

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

   if 2.94999999999999993e35 < beta

   1. Initial program 81.4%

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

   3. Simplified 58.0%

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

      1. times-frac 86.0%

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

      2. +-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in beta around inf 89.8%

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

      1. frac-times 84.3%

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

      2. *-rgt-identity 84.3%

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

      3. +-commutative 84.3%

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

   9. Applied egg-rr 84.3%

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

       1. associate-/l/ 89.8%

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

       2. +-commutative 89.8%

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

       3. +-commutative 89.8%

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

       4. +-commutative 89.8%

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

   11. Simplified 89.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.95 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5.5)
     (/
      (+
       0.16666666666666666
       (* alpha (+ 0.027777777777777776 (* alpha -0.05092592592592592))))
      t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5.5) {
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5.5d0) then
        tmp = (0.16666666666666666d0 + (alpha * (0.027777777777777776d0 + (alpha * (-0.05092592592592592d0))))) / t_0
    else
        tmp = ((1.0d0 + alpha) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5.5) {
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5.5:
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5.5)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * Float64(0.027777777777777776 + Float64(alpha * -0.05092592592592592)))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5.5)
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / t_0;
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.5], N[(N[(0.16666666666666666 + N[(alpha * N[(0.027777777777777776 + N[(alpha * -0.05092592592592592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5

   1. Initial program 99.8%

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

   3. Simplified 94.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around 0 98.7%

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

   10. Taylor expanded in alpha around 0 66.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + -0.05092592592592592 \cdot \alpha\right)}}{\alpha + \left(\beta + 2\right)} \]
   11. Step-by-step derivation

       1. *-commutative 66.5%

          \[\leadsto \frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \color{blue}{\alpha \cdot -0.05092592592592592}\right)}{\alpha + \left(\beta + 2\right)} \]

   12. Simplified 66.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}}{\alpha + \left(\beta + 2\right)} \]

   if 5.5 < beta

   1. Initial program 83.1%

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

   3. Simplified 61.8%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in beta around inf 88.1%

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

      1. frac-times 83.1%

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

      2. *-rgt-identity 83.1%

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

      3. +-commutative 83.1%

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

   9. Applied egg-rr 83.1%

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

       1. associate-/l/ 88.1%

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

       2. +-commutative 88.1%

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

       3. +-commutative 88.1%

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

       4. +-commutative 88.1%

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

   11. Simplified 88.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/
    (+
     0.16666666666666666
     (* alpha (+ 0.027777777777777776 (* alpha -0.05092592592592592))))
    (+ alpha (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / (alpha + (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (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) :: tmp
    if (beta <= 5.2d0) then
        tmp = (0.16666666666666666d0 + (alpha * (0.027777777777777776d0 + (alpha * (-0.05092592592592592d0))))) / (alpha + (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (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 tmp;
	if (beta <= 5.2) {
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / (alpha + (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / (alpha + (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * Float64(0.027777777777777776 + Float64(alpha * -0.05092592592592592)))) / Float64(alpha + Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2)
		tmp = (0.16666666666666666 + (alpha * (0.027777777777777776 + (alpha * -0.05092592592592592)))) / (alpha + (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (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_] := If[LessEqual[beta, 5.2], N[(N[(0.16666666666666666 + N[(alpha * N[(0.027777777777777776 + N[(alpha * -0.05092592592592592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   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 94.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around 0 98.7%

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

   10. Taylor expanded in alpha around 0 66.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + -0.05092592592592592 \cdot \alpha\right)}}{\alpha + \left(\beta + 2\right)} \]
   11. Step-by-step derivation

       1. *-commutative 66.5%

          \[\leadsto \frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \color{blue}{\alpha \cdot -0.05092592592592592}\right)}{\alpha + \left(\beta + 2\right)} \]

   12. Simplified 66.5%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}}{\alpha + \left(\beta + 2\right)} \]

   if 5.20000000000000018 < beta

   1. Initial program 83.1%

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

   3. Taylor expanded in beta around inf 88.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+35)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+35) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2d+35) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+35) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+35:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+35)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+35)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2e+35], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9999999999999999e35

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+l+ 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 84.6%

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

      1. +-commutative 84.6%

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

   7. Simplified 84.6%

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

   8. Taylor expanded in alpha around 0 69.1%

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

   if 1.9999999999999999e35 < beta

   1. Initial program 81.4%

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

   3. Simplified 58.0%

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

      1. times-frac 86.0%

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

      2. +-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in beta around inf 89.8%

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

      1. frac-times 84.3%

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

      2. *-rgt-identity 84.3%

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

      3. +-commutative 84.3%

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

   9. Applied egg-rr 84.3%

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

       1. associate-/l/ 89.8%

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

       2. +-commutative 89.8%

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

       3. +-commutative 89.8%

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

       4. +-commutative 89.8%

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

   11. Simplified 89.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.5)
   (/
    (+ 0.16666666666666666 (* alpha 0.027777777777777776))
    (+ alpha (+ beta 2.0)))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.5d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.027777777777777776d0)) / (alpha + (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.5:
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.5)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.027777777777777776)) / Float64(alpha + Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.5)
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / (alpha + (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.5], N[(N[(0.16666666666666666 + N[(alpha * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5

   1. Initial program 99.8%

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

   3. Simplified 94.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around 0 98.7%

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

   10. Taylor expanded in alpha around 0 67.2%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \alpha}}{\alpha + \left(\beta + 2\right)} \]
   11. Step-by-step derivation

       1. *-commutative 67.2%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\alpha \cdot 0.027777777777777776}}{\alpha + \left(\beta + 2\right)} \]

   12. Simplified 67.2%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot 0.027777777777777776}}{\alpha + \left(\beta + 2\right)} \]

   if 7.5 < beta

   1. Initial program 83.1%

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

   3. Simplified 61.8%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in beta around inf 88.1%

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

   8. Taylor expanded in beta around inf 87.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 6.1:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 6.1)
     (/ (+ 0.16666666666666666 (* alpha 0.027777777777777776)) t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6.1) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 6.1d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.027777777777777776d0)) / t_0
    else
        tmp = ((1.0d0 + alpha) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 6.1) {
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 6.1:
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 6.1)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.027777777777777776)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 6.1)
		tmp = (0.16666666666666666 + (alpha * 0.027777777777777776)) / t_0;
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.1], N[(N[(0.16666666666666666 + N[(alpha * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.0999999999999996

   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 94.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around 0 98.7%

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

   10. Taylor expanded in alpha around 0 67.2%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + 0.027777777777777776 \cdot \alpha}}{\alpha + \left(\beta + 2\right)} \]
   11. Step-by-step derivation

       1. *-commutative 67.2%

          \[\leadsto \frac{0.16666666666666666 + \color{blue}{\alpha \cdot 0.027777777777777776}}{\alpha + \left(\beta + 2\right)} \]

   12. Simplified 67.2%

       \[\leadsto \frac{\color{blue}{0.16666666666666666 + \alpha \cdot 0.027777777777777776}}{\alpha + \left(\beta + 2\right)} \]

   if 6.0999999999999996 < beta

   1. Initial program 83.1%

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

   3. Simplified 61.8%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in beta around inf 88.1%

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

      1. frac-times 83.1%

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

      2. *-rgt-identity 83.1%

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

      3. +-commutative 83.1%

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

   9. Applied egg-rr 83.1%

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

       1. associate-/l/ 88.1%

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

       2. +-commutative 88.1%

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

       3. +-commutative 88.1%

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

       4. +-commutative 88.1%

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

   11. Simplified 88.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.1:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8)
   (/ 0.16666666666666666 (+ beta 2.0))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.8d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.8:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.8)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.8], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.79999999999999982

   1. Initial program 99.8%

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

   3. Simplified 94.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around 0 98.7%

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

   10. Taylor expanded in alpha around 0 66.5%

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

   if 7.79999999999999982 < beta

   1. Initial program 83.1%

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

   3. Simplified 61.8%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in beta around inf 88.1%

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

   8. Taylor expanded in beta around inf 87.9%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = 1.0 / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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 94.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around 0 98.7%

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

   10. Taylor expanded in alpha around 0 66.5%

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

   if 6 < beta

   1. Initial program 83.1%

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

   3. Simplified 61.8%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in beta around inf 88.1%

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

   8. Taylor expanded in alpha around 0 75.7%

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

4. Final simplification 69.7%

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

Alternative 15: 91.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = (1.0 / beta) / (beta + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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 94.3%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in beta around 0 98.7%

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

   10. Taylor expanded in alpha around 0 66.5%

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

   if 6 < beta

   1. Initial program 83.1%

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

   3. Simplified 61.8%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in beta around inf 88.1%

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

   8. Taylor expanded in alpha around 0 75.7%

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

      1. associate-/r* 76.0%

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

      2. +-commutative 76.0%

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

   10. Simplified 76.0%

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

4. Final simplification 69.8%

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

Alternative 16: 12.2% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+l+ 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around inf 28.2%

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

   6. Taylor expanded in alpha around 0 14.1%

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

   7. Taylor expanded in beta around 0 14.1%

      \[\leadsto \color{blue}{0.16666666666666666} \]

   if 6 < beta

   1. Initial program 83.1%

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

   3. Simplified 61.8%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in beta around inf 88.1%

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

   8. Taylor expanded in alpha around inf 7.3%

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

4. Final simplification 11.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 83.1%

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

   1. times-frac 95.5%

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

   2. +-commutative 95.5%

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

6. Applied egg-rr 95.5%

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

   1. associate-*l/ 95.5%

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

   2. +-commutative 95.5%

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

   3. associate-+r+ 95.5%

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

   4. associate-/r* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in beta around 0 69.5%

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

10. Taylor expanded in alpha around 0 46.2%

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

11. Final simplification 46.2%

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

Alternative 18: 10.5% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.16666666666666666
end

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.16666666666666666

Derivation

1. Initial program 94.1%

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

   1. associate-/l/ 92.5%

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

   2. +-commutative 92.5%

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

   3. associate-+l+ 92.5%

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

   4. *-commutative 92.5%

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

   5. metadata-eval 92.5%

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

   6. associate-+l+ 92.5%

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

   7. metadata-eval 92.5%

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

   8. associate-+l+ 92.5%

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

   9. metadata-eval 92.5%

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

   10. metadata-eval 92.5%

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

   11. associate-+l+ 92.5%

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

3. Simplified 92.5%

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

5. Taylor expanded in beta around inf 47.1%

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

6. Taylor expanded in alpha around 0 35.3%

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

7. Taylor expanded in beta around 0 10.6%

   \[\leadsto \color{blue}{0.16666666666666666} \]

8. Final simplification 10.6%

   \[\leadsto 0.16666666666666666 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024069 
(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)))