Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 98.5%

Time: 15.2s

Alternatives: 13

Speedup: 3.2×

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: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.1e+38)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.1e+38) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.1d+38) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.1e+38) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.1e+38:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.1e+38)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.1e+38)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.1e+38], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.1000000000000001e38

   1. Initial program 99.3%

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

      1. associate-/l/ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r+ 99.3%

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

      5. associate-+l+ 99.3%

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

      6. distribute-rgt1-in 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. +-commutative 99.3%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 69.8%

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

      1. associate-*r/ 69.9%

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

      2. +-commutative 69.9%

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

      3. *-commutative 69.9%

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

   6. Applied egg-rr 69.9%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   if 5.1000000000000001e38 < beta

   1. Initial program 86.1%

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

      1. associate-/l/ 81.6%

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

      2. associate-+l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-+l+ 81.6%

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

      6. distribute-rgt1-in 81.6%

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

      7. *-rgt-identity 81.6%

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

      8. distribute-lft-out 81.6%

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

      9. +-commutative 81.6%

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

      10. associate-*r/ 93.3%

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

      11. associate-*r/ 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in alpha around 0 67.0%

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

      1. associate-*r/ 85.5%

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

      2. +-commutative 85.5%

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

      3. *-commutative 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. *-un-lft-identity 85.5%

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

      2. associate-+r+ 85.5%

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

      3. metadata-eval 85.5%

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

      4. associate-*r/ 76.5%

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

      5. +-commutative 76.5%

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

      6. metadata-eval 76.5%

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

      7. associate-+r+ 76.5%

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

   8. Applied egg-rr 76.5%

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

      1. *-lft-identity 76.5%

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

      2. associate-/r* 81.0%

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

      3. associate-/l* 89.4%

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

      4. associate-+r+ 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   10. Simplified 89.4%

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

   11. Taylor expanded in beta around inf 89.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 2: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 94.3%

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

   2. associate-+l+ 94.3%

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

   3. +-commutative 94.3%

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

   4. associate-+r+ 94.3%

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

   5. associate-+l+ 94.3%

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

   6. distribute-rgt1-in 94.3%

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

   7. *-rgt-identity 94.3%

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

   8. distribute-lft-out 94.3%

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

   9. +-commutative 94.3%

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

   10. associate-*r/ 98.0%

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

   11. associate-*r/ 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in alpha around 0 69.0%

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

   1. associate-*r/ 74.3%

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

   2. +-commutative 74.3%

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

   3. *-commutative 74.3%

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

6. Applied egg-rr 74.3%

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

   1. *-un-lft-identity 74.3%

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

   2. associate-+r+ 74.3%

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

   3. metadata-eval 74.3%

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

   4. associate-*r/ 71.7%

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

   5. +-commutative 71.7%

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

   6. metadata-eval 71.7%

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

   7. associate-+r+ 71.7%

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

8. Applied egg-rr 71.7%

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

   1. *-lft-identity 71.7%

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

   2. associate-/r* 73.0%

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

   3. associate-/l* 75.3%

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

   4. associate-+r+ 75.3%

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

   5. +-commutative 75.3%

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

   6. +-commutative 75.3%

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

10. Simplified 75.3%

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

11. Final simplification 75.3%

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

Alternative 3: 98.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta \cdot \left(\beta + 5\right) + 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.2e+15)
   (/ (+ beta 1.0) (* (+ beta 2.0) (+ (* beta (+ beta 5.0)) 6.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2e+15) {
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta * (beta + 5.0)) + 6.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.2d+15) then
        tmp = (beta + 1.0d0) / ((beta + 2.0d0) * ((beta * (beta + 5.0d0)) + 6.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2e+15) {
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta * (beta + 5.0)) + 6.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.2e+15:
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta * (beta + 5.0)) + 6.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.2e+15)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(Float64(beta * Float64(beta + 5.0)) + 6.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.2e+15)
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta * (beta + 5.0)) + 6.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.2e+15], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.2e15

   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. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.3%

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

   5. Taylor expanded in beta around 0 70.3%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{6 + \left(5 \cdot \beta + {\beta}^{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 70.3%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{6 + \color{blue}{\left({\beta}^{2} + 5 \cdot \beta\right)}} \]

      2. unpow2 70.3%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{6 + \left(\color{blue}{\beta \cdot \beta} + 5 \cdot \beta\right)} \]

      3. distribute-rgt-out 70.3%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{6 + \color{blue}{\beta \cdot \left(\beta + 5\right)}} \]

   7. Simplified 70.3%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{6 + \beta \cdot \left(\beta + 5\right)}} \]

   8. Taylor expanded in alpha around 0 70.4%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta \cdot \left(\beta + 5\right) + 6\right)}} \]

   if 7.2e15 < beta

   1. Initial program 86.0%

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

      1. associate-/l/ 81.9%

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

      2. associate-+l+ 81.9%

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

      3. +-commutative 81.9%

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

      4. associate-+r+ 81.9%

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

      5. associate-+l+ 81.9%

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

      6. distribute-rgt1-in 81.9%

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

      7. *-rgt-identity 81.9%

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

      8. distribute-lft-out 81.9%

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

      9. +-commutative 81.9%

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

      10. associate-*r/ 93.9%

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

      11. associate-*r/ 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in alpha around 0 66.2%

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

      1. associate-*r/ 83.1%

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

      2. +-commutative 83.1%

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

      3. *-commutative 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. associate-+r+ 83.1%

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

      3. metadata-eval 83.1%

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

      4. associate-*r/ 74.9%

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

      5. +-commutative 74.9%

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

      6. metadata-eval 74.9%

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

      7. associate-+r+ 74.9%

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

   8. Applied egg-rr 74.9%

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

      1. *-lft-identity 74.9%

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

      2. associate-/r* 78.9%

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

      3. associate-/l* 86.6%

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

      4. associate-+r+ 86.6%

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

      5. +-commutative 86.6%

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

      6. +-commutative 86.6%

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

   10. Simplified 86.6%

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

   11. Taylor expanded in beta around inf 86.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta \cdot \left(\beta + 5\right) + 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 4: 97.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{\frac{0.5 + \alpha \cdot 0.5}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.3)
   (/ (/ (+ 0.5 (* alpha 0.5)) (+ alpha 2.0)) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = ((0.5 + (alpha * 0.5)) / (alpha + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.3d0) then
        tmp = ((0.5d0 + (alpha * 0.5d0)) / (alpha + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = ((0.5 + (alpha * 0.5)) / (alpha + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.3:
		tmp = ((0.5 + (alpha * 0.5)) / (alpha + 2.0)) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.3)
		tmp = Float64(Float64(Float64(0.5 + Float64(alpha * 0.5)) / Float64(alpha + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.3)
		tmp = ((0.5 + (alpha * 0.5)) / (alpha + 2.0)) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.3], N[(N[(N[(0.5 + N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.29999999999999982

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. associate-*r/ 70.1%

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

      2. +-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-+r+ 70.1%

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

      3. metadata-eval 70.1%

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

      4. associate-*r/ 70.1%

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

      5. +-commutative 70.1%

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

      6. metadata-eval 70.1%

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

      7. associate-+r+ 70.1%

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

   8. Applied egg-rr 70.1%

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

      1. *-lft-identity 70.1%

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

      2. associate-/r* 70.1%

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

      3. associate-/l* 70.1%

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

      4. associate-+r+ 70.1%

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

      5. +-commutative 70.1%

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

      6. +-commutative 70.1%

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

   10. Simplified 70.1%

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

   11. Taylor expanded in beta around 0 69.2%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1 + \alpha}{2 + \alpha}}}{\beta + 3} \]
   12. Step-by-step derivation

       1. associate-*r/ 69.2%

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

       2. distribute-lft-in 69.2%

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

       3. metadata-eval 69.2%

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

       4. +-commutative 69.2%

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

   13. Simplified 69.2%

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

   if 4.29999999999999982 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.2%

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

      2. associate-+l+ 82.2%

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

      3. +-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. associate-+l+ 82.2%

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

      6. distribute-rgt1-in 82.2%

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

      7. *-rgt-identity 82.2%

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

      8. distribute-lft-out 82.2%

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

      9. +-commutative 82.2%

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

      10. associate-*r/ 94.0%

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

      11. associate-*r/ 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in alpha around 0 66.6%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. *-un-lft-identity 83.3%

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

      2. associate-+r+ 83.3%

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

      3. metadata-eval 83.3%

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

      4. associate-*r/ 75.2%

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

      5. +-commutative 75.2%

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

      6. metadata-eval 75.2%

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

      7. associate-+r+ 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. *-lft-identity 75.2%

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

      2. associate-/r* 79.2%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. +-commutative 86.7%

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

      6. +-commutative 86.7%

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

   10. Simplified 86.7%

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

   11. Taylor expanded in beta around inf 86.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{\frac{0.5 + \alpha \cdot 0.5}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 5: 93.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.65:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \alpha}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.65)
   (* 0.16666666666666666 (/ (+ 1.0 alpha) (+ alpha 2.0)))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.65) {
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.65d0) then
        tmp = 0.16666666666666666d0 * ((1.0d0 + alpha) / (alpha + 2.0d0))
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.65) {
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.65:
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0))
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.65)
		tmp = Float64(0.16666666666666666 * Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.65)
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.65], N[(0.16666666666666666 * N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.64999999999999991

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

   5. Taylor expanded in beta around 0 68.2%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]

   if 3.64999999999999991 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.2%

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

      2. associate-+l+ 82.2%

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

      3. +-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. associate-+l+ 82.2%

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

      6. distribute-rgt1-in 82.2%

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

      7. *-rgt-identity 82.2%

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

      8. distribute-lft-out 82.2%

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

      9. +-commutative 82.2%

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

      10. associate-*r/ 94.0%

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

      11. associate-*r/ 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in alpha around 0 66.6%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in beta around inf 83.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 83.1%

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

   9. Simplified 83.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.65:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \alpha}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 6: 93.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.65:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.65)
   (/ (* (+ 1.0 alpha) 0.16666666666666666) (+ alpha 2.0))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.65) {
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.65d0) then
        tmp = ((1.0d0 + alpha) * 0.16666666666666666d0) / (alpha + 2.0d0)
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.65) {
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.65:
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.65)
		tmp = Float64(Float64(Float64(1.0 + alpha) * 0.16666666666666666) / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.65)
		tmp = ((1.0 + alpha) * 0.16666666666666666) / (alpha + 2.0);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.65], N[(N[(N[(1.0 + alpha), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.64999999999999991

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. associate-*r/ 70.1%

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

      2. +-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in beta around 0 68.2%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   8. Step-by-step derivation

      1. associate-*r/ 68.2%

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

   9. Simplified 68.2%

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

   if 3.64999999999999991 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.2%

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

      2. associate-+l+ 82.2%

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

      3. +-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. associate-+l+ 82.2%

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

      6. distribute-rgt1-in 82.2%

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

      7. *-rgt-identity 82.2%

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

      8. distribute-lft-out 82.2%

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

      9. +-commutative 82.2%

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

      10. associate-*r/ 94.0%

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

      11. associate-*r/ 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in alpha around 0 66.6%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in beta around inf 83.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 83.1%

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

   9. Simplified 83.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.65:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 7: 93.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6)
   (/ (+ 0.16666666666666666 (* alpha 0.16666666666666666)) (+ alpha 2.0))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.16666666666666666d0)) / (alpha + 2.0d0)
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.16666666666666666)) / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.6], N[(N[(0.16666666666666666 + N[(alpha * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. associate-*r/ 70.1%

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

      2. +-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-+r+ 70.1%

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

      3. metadata-eval 70.1%

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

      4. associate-*r/ 70.1%

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

      5. +-commutative 70.1%

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

      6. metadata-eval 70.1%

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

      7. associate-+r+ 70.1%

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

   8. Applied egg-rr 70.1%

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

      1. *-lft-identity 70.1%

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

      2. associate-/r* 70.1%

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

      3. associate-/l* 70.1%

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

      4. associate-+r+ 70.1%

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

      5. +-commutative 70.1%

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

      6. +-commutative 70.1%

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

   10. Simplified 70.1%

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

   11. Taylor expanded in beta around 0 68.2%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   12. Step-by-step derivation

       1. associate-*r/ 68.2%

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

       2. distribute-lft-in 68.2%

          \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot 1 + 0.16666666666666666 \cdot \alpha}}{2 + \alpha} \]

       3. metadata-eval 68.2%

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

       4. +-commutative 68.2%

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

   13. Simplified 68.2%

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

   if 3.60000000000000009 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.2%

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

      2. associate-+l+ 82.2%

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

      3. +-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. associate-+l+ 82.2%

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

      6. distribute-rgt1-in 82.2%

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

      7. *-rgt-identity 82.2%

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

      8. distribute-lft-out 82.2%

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

      9. +-commutative 82.2%

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

      10. associate-*r/ 94.0%

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

      11. associate-*r/ 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in alpha around 0 66.6%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in beta around inf 83.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 83.1%

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

   9. Simplified 83.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 8: 96.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ (+ 0.16666666666666666 (* alpha 0.16666666666666666)) (+ alpha 2.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = (0.16666666666666666d0 + (alpha * 0.16666666666666666d0)) / (alpha + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(Float64(0.16666666666666666 + Float64(alpha * 0.16666666666666666)) / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = (0.16666666666666666 + (alpha * 0.16666666666666666)) / (alpha + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(N[(0.16666666666666666 + N[(alpha * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. associate-*r/ 70.1%

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

      2. +-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-+r+ 70.1%

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

      3. metadata-eval 70.1%

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

      4. associate-*r/ 70.1%

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

      5. +-commutative 70.1%

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

      6. metadata-eval 70.1%

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

      7. associate-+r+ 70.1%

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

   8. Applied egg-rr 70.1%

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

      1. *-lft-identity 70.1%

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

      2. associate-/r* 70.1%

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

      3. associate-/l* 70.1%

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

      4. associate-+r+ 70.1%

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

      5. +-commutative 70.1%

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

      6. +-commutative 70.1%

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

   10. Simplified 70.1%

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

   11. Taylor expanded in beta around 0 68.2%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   12. Step-by-step derivation

       1. associate-*r/ 68.2%

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

       2. distribute-lft-in 68.2%

          \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot 1 + 0.16666666666666666 \cdot \alpha}}{2 + \alpha} \]

       3. metadata-eval 68.2%

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

       4. +-commutative 68.2%

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

   13. Simplified 68.2%

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

   if 2.2999999999999998 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.2%

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

      2. associate-+l+ 82.2%

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

      3. +-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. associate-+l+ 82.2%

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

      6. distribute-rgt1-in 82.2%

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

      7. *-rgt-identity 82.2%

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

      8. distribute-lft-out 82.2%

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

      9. +-commutative 82.2%

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

      10. associate-*r/ 94.0%

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

      11. associate-*r/ 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in alpha around 0 66.6%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. *-un-lft-identity 83.3%

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

      2. associate-+r+ 83.3%

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

      3. metadata-eval 83.3%

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

      4. associate-*r/ 75.2%

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

      5. +-commutative 75.2%

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

      6. metadata-eval 75.2%

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

      7. associate-+r+ 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. *-lft-identity 75.2%

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

      2. associate-/r* 79.2%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. +-commutative 86.7%

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

      6. +-commutative 86.7%

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

   10. Simplified 86.7%

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

   11. Taylor expanded in beta around inf 86.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 9: 60.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4)
   (* (+ 1.0 alpha) 0.16666666666666666)
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = (1.0d0 + alpha) * 0.16666666666666666d0
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = (1.0 + alpha) * 0.16666666666666666
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(Float64(1.0 + alpha) * 0.16666666666666666);
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = (1.0 + alpha) * 0.16666666666666666;
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.4], N[(N[(1.0 + alpha), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. associate-*r/ 70.1%

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

      2. +-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in beta around inf 14.1%

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

   8. Taylor expanded in beta around 0 14.1%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(1 + \alpha\right)} \]

   if 2.39999999999999991 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.2%

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

      2. associate-+l+ 82.2%

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

      3. +-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. associate-+l+ 82.2%

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

      6. distribute-rgt1-in 82.2%

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

      7. *-rgt-identity 82.2%

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

      8. distribute-lft-out 82.2%

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

      9. +-commutative 82.2%

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

      10. associate-*r/ 94.0%

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

      11. associate-*r/ 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in alpha around 0 66.6%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in beta around inf 83.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 83.1%

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

   9. Simplified 83.1%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 10: 12.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 0.49:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.08333333333333333}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 0.49)
   (* (+ 1.0 alpha) 0.16666666666666666)
   (/ 0.08333333333333333 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.49) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = 0.08333333333333333 / 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 <= 0.49d0) then
        tmp = (1.0d0 + alpha) * 0.16666666666666666d0
    else
        tmp = 0.08333333333333333d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.49) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = 0.08333333333333333 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 0.49:
		tmp = (1.0 + alpha) * 0.16666666666666666
	else:
		tmp = 0.08333333333333333 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 0.49)
		tmp = Float64(Float64(1.0 + alpha) * 0.16666666666666666);
	else
		tmp = Float64(0.08333333333333333 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 0.49)
		tmp = (1.0 + alpha) * 0.16666666666666666;
	else
		tmp = 0.08333333333333333 / 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, 0.49], N[(N[(1.0 + alpha), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(0.08333333333333333 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 0.48999999999999999

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. associate-*r/ 70.1%

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

      2. +-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in beta around inf 14.1%

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

   8. Taylor expanded in beta around 0 14.1%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(1 + \alpha\right)} \]

   if 0.48999999999999999 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.1%

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

      2. associate-/l/ 61.3%

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

      3. associate-+l+ 61.3%

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

      4. +-commutative 61.3%

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

      5. associate-+r+ 61.3%

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

      6. associate-+l+ 61.3%

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

      7. distribute-rgt1-in 61.3%

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

      8. *-rgt-identity 61.3%

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

      9. distribute-lft-out 61.3%

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

      10. +-commutative 61.3%

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

      11. times-frac 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in beta around -inf 87.1%

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

      1. unpow2 87.1%

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

      2. associate-*r* 87.1%

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

      3. *-commutative 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in beta around 0 20.9%

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

      1. associate-*r/ 20.9%

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

      2. associate-*r* 20.9%

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

      3. times-frac 17.5%

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

      4. +-commutative 17.5%

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

   9. Simplified 17.5%

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

   10. Taylor expanded in alpha around 0 6.6%

       \[\leadsto \color{blue}{\frac{0.08333333333333333}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 0.49:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.08333333333333333}{\beta}\\ \end{array} \]

Alternative 11: 14.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8e+113)
   (* (+ 1.0 alpha) 0.16666666666666666)
   (/ 0.5 (* beta alpha))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+113) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = 0.5 / (beta * alpha);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.8d+113) then
        tmp = (1.0d0 + alpha) * 0.16666666666666666d0
    else
        tmp = 0.5d0 / (beta * alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+113) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = 0.5 / (beta * alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8e+113:
		tmp = (1.0 + alpha) * 0.16666666666666666
	else:
		tmp = 0.5 / (beta * alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8e+113)
		tmp = Float64(Float64(1.0 + alpha) * 0.16666666666666666);
	else
		tmp = Float64(0.5 / Float64(beta * alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8e+113)
		tmp = (1.0 + alpha) * 0.16666666666666666;
	else
		tmp = 0.5 / (beta * alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8e+113], N[(N[(1.0 + alpha), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(0.5 / N[(beta * alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999966e113

   1. Initial program 99.3%

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

      1. associate-/l/ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r+ 99.3%

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

      5. associate-+l+ 99.3%

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

      6. distribute-rgt1-in 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. +-commutative 99.3%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in alpha around 0 69.9%

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

      1. associate-*r/ 71.5%

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

      2. +-commutative 71.5%

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

      3. *-commutative 71.5%

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

   6. Applied egg-rr 71.5%

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

   7. Taylor expanded in beta around inf 23.3%

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

   8. Taylor expanded in beta around 0 12.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(1 + \alpha\right)} \]

   if 4.79999999999999966e113 < beta

   1. Initial program 80.5%

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

      1. associate-/l/ 74.3%

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

      2. associate-/l/ 52.7%

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

      3. associate-+l+ 52.7%

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

      4. +-commutative 52.7%

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

      5. associate-+r+ 52.7%

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

      6. associate-+l+ 52.7%

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

      7. distribute-rgt1-in 52.7%

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

      8. *-rgt-identity 52.7%

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

      9. distribute-lft-out 52.7%

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

      10. +-commutative 52.7%

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

      11. times-frac 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in beta around -inf 89.1%

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

      1. unpow2 89.1%

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

      2. associate-*r* 89.1%

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

      3. *-commutative 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in beta around 0 21.4%

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

      1. associate-*r/ 21.4%

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

      2. associate-*r* 21.4%

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

      3. times-frac 21.5%

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

      4. +-commutative 21.5%

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

   9. Simplified 21.5%

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

   10. Taylor expanded in alpha around inf 19.6%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta \cdot \alpha}\\ \end{array} \]

Alternative 12: 57.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (* (+ 1.0 alpha) 0.16666666666666666)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = (1.0d0 + alpha) * 0.16666666666666666d0
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = (1.0 + alpha) * 0.16666666666666666;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = (1.0 + alpha) * 0.16666666666666666
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(Float64(1.0 + alpha) * 0.16666666666666666);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = (1.0 + alpha) * 0.16666666666666666;
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(N[(1.0 + alpha), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in alpha around 0 70.1%

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

      1. associate-*r/ 70.1%

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

      2. +-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in beta around inf 14.1%

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

   8. Taylor expanded in beta around 0 14.1%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(1 + \alpha\right)} \]

   if 2.2999999999999998 < beta

   1. Initial program 86.2%

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

      1. associate-/l/ 82.2%

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

      2. associate-/r* 61.3%

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

      3. associate-+l+ 61.3%

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

      4. +-commutative 61.3%

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

      5. associate-+r+ 61.3%

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

      6. associate-+l+ 61.3%

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

      7. distribute-rgt1-in 61.3%

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

      8. *-rgt-identity 61.3%

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

      9. distribute-lft-out 61.3%

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

      10. *-commutative 61.3%

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

      11. metadata-eval 61.3%

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

      12. associate-+l+ 61.3%

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

      13. +-commutative 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in beta around inf 56.5%

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

   5. Taylor expanded in alpha around 0 66.6%

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

   6. Taylor expanded in beta around inf 79.7%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 79.7%

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

   8. Simplified 79.7%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\left(1 + \alpha\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 13: 6.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.08333333333333333}{\beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333 / beta;
}

Fortran

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

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.08333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.08333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 94.3%

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

   2. associate-/l/ 83.1%

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

   3. associate-+l+ 83.1%

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

   4. +-commutative 83.1%

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

   5. associate-+r+ 83.1%

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

   6. associate-+l+ 83.1%

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

   7. distribute-rgt1-in 83.1%

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

   8. *-rgt-identity 83.1%

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

   9. distribute-lft-out 83.1%

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

   10. +-commutative 83.1%

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

   11. times-frac 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in beta around -inf 29.3%

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

   1. unpow2 29.3%

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

   2. associate-*r* 29.3%

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

   3. *-commutative 29.3%

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

6. Simplified 29.3%

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

7. Taylor expanded in beta around 0 16.9%

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

   1. associate-*r/ 16.9%

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

   2. associate-*r* 13.5%

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

   3. times-frac 7.6%

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

   4. +-commutative 7.6%

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

9. Simplified 7.6%

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

10. Taylor expanded in alpha around 0 4.0%

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

11. Final simplification 4.0%

    \[\leadsto \frac{0.08333333333333333}{\beta} \]

Reproduce

herbie shell --seed 2023255 
(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)))