Octave 3.8, jcobi/3

Percentage Accurate: 93.9% → 99.8%

Time: 19.4s

Alternatives: 15

Speedup: 2.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. metadata-eval 94.0%

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

   2. div-inv 94.0%

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

3. Applied egg-rr 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-*l/ 94.0%

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

   3. *-lft-identity 94.0%

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

   4. *-commutative 94.0%

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

   5. /-rgt-identity 94.0%

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

   6. associate-*r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   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/ 99.9%

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

      2. associate-/r* 95.2%

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

      3. +-commutative 95.2%

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

      4. +-commutative 95.2%

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

      5. *-commutative 95.2%

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

      6. associate-+l+ 95.2%

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

      7. associate-+r+ 95.2%

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

      8. *-commutative 95.2%

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

      9. distribute-rgt1-in 95.2%

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

      10. +-commutative 95.2%

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

      11. *-commutative 95.2%

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

      12. distribute-rgt1-in 95.2%

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

      13. +-commutative 95.2%

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

      14. metadata-eval 95.2%

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

      15. associate-+l+ 95.2%

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

      16. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in beta around 0 94.3%

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

   if 3.7999999999999998 < beta

   1. Initial program 81.5%

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

      1. metadata-eval 81.5%

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

      2. div-inv 81.5%

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

   3. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l/ 81.5%

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

      3. *-lft-identity 81.5%

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

      4. *-commutative 81.5%

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

      5. /-rgt-identity 81.5%

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

      6. associate-*r/ 99.7%

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

      7. /-rgt-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. associate-+l+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-/l/ 87.1%

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

      9. +-commutative 87.1%

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

      10. *-commutative 87.1%

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

      11. associate-+r+ 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-+r+ 87.1%

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

      14. metadata-eval 87.1%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in beta around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. distribute-lft-in 85.4%

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

      3. metadata-eval 85.4%

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

      4. neg-mul-1 85.4%

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

      5. unsub-neg 85.4%

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

   10. Simplified 85.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. metadata-eval 94.0%

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

   2. div-inv 94.0%

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

3. Applied egg-rr 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-*l/ 94.0%

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

   3. *-lft-identity 94.0%

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

   4. *-commutative 94.0%

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

   5. /-rgt-identity 94.0%

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

   6. associate-*r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

5. Simplified 99.8%

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

   1. metadata-eval 99.8%

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

   2. associate-+l+ 99.8%

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

   3. metadata-eval 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.8%

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

   7. metadata-eval 99.8%

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

   8. associate-/l/ 95.8%

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

   9. +-commutative 95.8%

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

   10. *-commutative 95.8%

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

   11. associate-+r+ 95.8%

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

   12. +-commutative 95.8%

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

   13. associate-+r+ 95.8%

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

   14. metadata-eval 95.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.7:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.7)
   (/
    (* (+ 1.0 beta) (+ 1.0 alpha))
    (* (+ alpha (+ beta 2.0)) (* (+ alpha 2.0) (+ alpha 3.0))))
   (*
    (+ 1.0 (/ (- -1.0 alpha) beta))
    (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.7) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.7d0) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / ((alpha + (beta + 2.0d0)) * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = (1.0d0 + (((-1.0d0) - alpha) / beta)) * (((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.7) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.7:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.7)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) * Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.7)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((alpha + (beta + 2.0)) * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.7], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.70000000000000018

   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/ 99.9%

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

      2. associate-/r* 95.2%

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

      3. +-commutative 95.2%

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

      4. +-commutative 95.2%

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

      5. *-commutative 95.2%

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

      6. associate-+l+ 95.2%

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

      7. associate-+r+ 95.2%

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

      8. *-commutative 95.2%

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

      9. distribute-rgt1-in 95.2%

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

      10. +-commutative 95.2%

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

      11. *-commutative 95.2%

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

      12. distribute-rgt1-in 95.2%

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

      13. +-commutative 95.2%

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

      14. metadata-eval 95.2%

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

      15. associate-+l+ 95.2%

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

      16. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in beta around 0 94.3%

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

   if 6.70000000000000018 < beta

   1. Initial program 81.5%

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

      1. metadata-eval 81.5%

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

      2. div-inv 81.5%

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

   3. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l/ 81.5%

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

      3. *-lft-identity 81.5%

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

      4. *-commutative 81.5%

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

      5. /-rgt-identity 81.5%

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

      6. associate-*r/ 99.7%

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

      7. /-rgt-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. associate-+l+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-/l/ 87.1%

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

      9. +-commutative 87.1%

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

      10. *-commutative 87.1%

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

      11. associate-+r+ 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-+r+ 87.1%

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

      14. metadata-eval 87.1%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in beta around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. distribute-lft-in 85.4%

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

      3. metadata-eval 85.4%

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

      4. neg-mul-1 85.4%

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

      5. unsub-neg 85.4%

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

   10. Simplified 85.4%

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

   11. Taylor expanded in beta around inf 84.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.7:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 14.5:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t_0 \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\beta + \left(\alpha + 3\right)} \cdot \left(1 - \frac{\alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 14.5)
     (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (* t_0 (* (+ alpha 2.0) (+ alpha 3.0))))
     (*
      (/ (/ (+ 1.0 alpha) t_0) (+ beta (+ alpha 3.0)))
      (- 1.0 (/ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 14.5) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (((1.0 + alpha) / t_0) / (beta + (alpha + 3.0))) * (1.0 - (alpha / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 14.5d0) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / (t_0 * ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = (((1.0d0 + alpha) / t_0) / (beta + (alpha + 3.0d0))) * (1.0d0 - (alpha / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 14.5) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (((1.0 + alpha) / t_0) / (beta + (alpha + 3.0))) * (1.0 - (alpha / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 14.5:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = (((1.0 + alpha) / t_0) / (beta + (alpha + 3.0))) * (1.0 - (alpha / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 14.5)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(t_0 * Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(beta + Float64(alpha + 3.0))) * Float64(1.0 - Float64(alpha / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 14.5)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_0 * ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = (((1.0 + alpha) / t_0) / (beta + (alpha + 3.0))) * (1.0 - (alpha / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 14.5], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 14.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.9%

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

      2. associate-/r* 95.2%

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

      3. +-commutative 95.2%

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

      4. +-commutative 95.2%

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

      5. *-commutative 95.2%

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

      6. associate-+l+ 95.2%

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

      7. associate-+r+ 95.2%

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

      8. *-commutative 95.2%

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

      9. distribute-rgt1-in 95.2%

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

      10. +-commutative 95.2%

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

      11. *-commutative 95.2%

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

      12. distribute-rgt1-in 95.2%

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

      13. +-commutative 95.2%

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

      14. metadata-eval 95.2%

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

      15. associate-+l+ 95.2%

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

      16. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in beta around 0 94.3%

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

   if 14.5 < beta

   1. Initial program 81.5%

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

      1. metadata-eval 81.5%

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

      2. div-inv 81.5%

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

   3. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l/ 81.5%

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

      3. *-lft-identity 81.5%

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

      4. *-commutative 81.5%

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

      5. /-rgt-identity 81.5%

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

      6. associate-*r/ 99.7%

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

      7. /-rgt-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. associate-+l+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-/l/ 87.1%

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

      9. +-commutative 87.1%

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

      10. *-commutative 87.1%

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

      11. associate-+r+ 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-+r+ 87.1%

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

      14. metadata-eval 87.1%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in beta around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. distribute-lft-in 85.4%

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

      3. metadata-eval 85.4%

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

      4. neg-mul-1 85.4%

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

      5. unsub-neg 85.4%

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

   10. Simplified 85.4%

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

   11. Taylor expanded in alpha around inf 84.8%

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

       1. mul-1-neg 84.8%

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

       2. distribute-neg-frac 84.8%

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

   13. Simplified 84.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 14.5:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + \left(\alpha + 3\right)} \cdot \left(1 - \frac{\alpha}{\beta}\right)\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7e+16)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (*
    (+ 1.0 (/ (- -1.0 alpha) beta))
    (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 2.7d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = (1.0d0 + (((-1.0d0) - alpha) / beta)) * (((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.7e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) * Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 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, 2.7e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7e16

   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/ 99.9%

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

      2. *-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in alpha around 0 87.7%

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

      1. +-commutative 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in alpha around 0 71.2%

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

      1. +-commutative 71.2%

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

   9. Simplified 71.2%

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

   if 2.7e16 < beta

   1. Initial program 81.1%

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

      1. metadata-eval 81.1%

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

      2. div-inv 81.1%

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

   3. Applied egg-rr 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*l/ 81.1%

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

      3. *-lft-identity 81.1%

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

      4. *-commutative 81.1%

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

      5. /-rgt-identity 81.1%

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

      6. associate-*r/ 99.8%

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

      7. /-rgt-identity 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. metadata-eval 99.8%

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

      2. associate-+l+ 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l/ 86.8%

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

      9. +-commutative 86.8%

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

      10. *-commutative 86.8%

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

      11. associate-+r+ 86.8%

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

      12. +-commutative 86.8%

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

      13. associate-+r+ 86.8%

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

      14. metadata-eval 86.8%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in beta around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. distribute-lft-in 85.4%

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

      3. metadata-eval 85.4%

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

      4. neg-mul-1 85.4%

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

      5. unsub-neg 85.4%

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

   10. Simplified 85.4%

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

   11. Taylor expanded in beta around inf 85.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 7: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 4.2)
     (/ (/ (+ 1.0 alpha) (+ alpha 3.0)) (* t_0 t_0))
     (*
      (+ 1.0 (/ (- -1.0 alpha) beta))
      (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.2) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (t_0 * t_0);
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 4.2d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 3.0d0)) / (t_0 * t_0)
    else
        tmp = (1.0d0 + (((-1.0d0) - alpha) / beta)) * (((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.2) {
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (t_0 * t_0);
	} else {
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 4.2:
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (t_0 * t_0)
	else:
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 3.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 3.0)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)) * Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = ((1.0 + alpha) / (alpha + 3.0)) / (t_0 * t_0);
	else
		tmp = (1.0 + ((-1.0 - alpha) / beta)) * (((1.0 + alpha) / beta) / (beta + (alpha + 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.2], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-/l/ 95.2%

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

      3. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in beta around 0 98.9%

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

   if 4.20000000000000018 < beta

   1. Initial program 81.5%

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

      1. metadata-eval 81.5%

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

      2. div-inv 81.5%

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

   3. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l/ 81.5%

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

      3. *-lft-identity 81.5%

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

      4. *-commutative 81.5%

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

      5. /-rgt-identity 81.5%

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

      6. associate-*r/ 99.7%

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

      7. /-rgt-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. associate-+l+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-/l/ 87.1%

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

      9. +-commutative 87.1%

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

      10. *-commutative 87.1%

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

      11. associate-+r+ 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-+r+ 87.1%

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

      14. metadata-eval 87.1%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in beta around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. distribute-lft-in 85.4%

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

      3. metadata-eval 85.4%

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

      4. neg-mul-1 85.4%

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

      5. unsub-neg 85.4%

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

   10. Simplified 85.4%

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

   11. Taylor expanded in beta around inf 84.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 - \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 8: 96.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1

   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/ 99.9%

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

      2. associate-/r* 95.2%

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

      3. +-commutative 95.2%

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

      4. +-commutative 95.2%

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

      5. *-commutative 95.2%

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

      6. associate-+l+ 95.2%

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

      7. associate-+r+ 95.2%

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

      8. *-commutative 95.2%

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

      9. distribute-rgt1-in 95.2%

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

      10. +-commutative 95.2%

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

      11. *-commutative 95.2%

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

      12. distribute-rgt1-in 95.2%

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

      13. +-commutative 95.2%

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

      14. metadata-eval 95.2%

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

      15. associate-+l+ 95.2%

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

      16. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in beta around 0 94.3%

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

   5. Taylor expanded in alpha around 0 69.9%

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

   if 1 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 77.6%

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

      2. associate-/l/ 60.4%

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

      3. associate-/r* 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in beta around inf 83.8%

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

      1. associate-+r+ 83.8%

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

      2. metadata-eval 83.8%

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

      3. associate-+r+ 83.8%

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

      4. metadata-eval 83.8%

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

      5. associate-/r* 85.3%

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

      6. div-inv 85.2%

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

      7. metadata-eval 85.2%

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

      8. associate-+r+ 85.2%

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

      9. metadata-eval 85.2%

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

      10. associate-+r+ 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. associate-*r/ 85.3%

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

      2. *-rgt-identity 85.3%

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

      3. +-commutative 85.3%

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

      4. +-commutative 85.3%

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

   8. Simplified 85.3%

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

4. Final simplification 74.8%

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

Alternative 9: 98.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.5e+15)
     (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
     (/ (/ (+ 1.0 alpha) t_0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.5e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3.5d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.5e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.5e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / t_0) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.5e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.5e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.5e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5e15

   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/ 99.9%

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

      2. *-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in alpha around 0 87.7%

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

      1. +-commutative 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in alpha around 0 71.2%

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

      1. +-commutative 71.2%

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

   9. Simplified 71.2%

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

   if 3.5e15 < beta

   1. Initial program 81.1%

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

      1. associate-/l/ 77.0%

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

      2. associate-/l/ 59.4%

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

      3. associate-/r* 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in beta around inf 84.5%

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

      1. associate-+r+ 84.5%

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

      2. metadata-eval 84.5%

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

      3. associate-+r+ 84.5%

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

      4. metadata-eval 84.5%

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

      5. associate-/r* 86.0%

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

      6. div-inv 85.9%

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

      7. metadata-eval 85.9%

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

      8. associate-+r+ 85.9%

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

      9. metadata-eval 85.9%

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

      10. associate-+r+ 85.9%

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

   6. Applied egg-rr 85.9%

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

      1. associate-*r/ 86.0%

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

      2. *-rgt-identity 86.0%

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

      3. +-commutative 86.0%

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

      4. +-commutative 86.0%

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

   8. Simplified 86.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]

Alternative 10: 96.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   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/ 99.9%

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

      2. associate-/r* 95.2%

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

      3. +-commutative 95.2%

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

      4. +-commutative 95.2%

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

      5. *-commutative 95.2%

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

      6. associate-+l+ 95.2%

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

      7. associate-+r+ 95.2%

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

      8. *-commutative 95.2%

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

      9. distribute-rgt1-in 95.2%

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

      10. +-commutative 95.2%

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

      11. *-commutative 95.2%

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

      12. distribute-rgt1-in 95.2%

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

      13. +-commutative 95.2%

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

      14. metadata-eval 95.2%

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

      15. associate-+l+ 95.2%

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

      16. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in beta around 0 94.3%

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

   5. Taylor expanded in alpha around 0 69.9%

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

   if 2 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 77.6%

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

      2. associate-/l/ 60.4%

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

      3. associate-/r* 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in beta around inf 83.8%

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

      1. associate-+r+ 83.8%

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

      2. metadata-eval 83.8%

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

      3. associate-+r+ 83.8%

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

      4. metadata-eval 83.8%

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

      5. associate-/r* 85.3%

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

      6. div-inv 85.2%

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

      7. metadata-eval 85.2%

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

      8. associate-+r+ 85.2%

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

      9. metadata-eval 85.2%

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

      10. associate-+r+ 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in beta around inf 84.8%

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

4. Final simplification 74.6%

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

Alternative 11: 96.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \beta}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.75)
   (* 0.16666666666666666 (/ (+ 1.0 beta) (+ beta 2.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.75) {
		tmp = 0.16666666666666666 * ((1.0 + beta) / (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (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 <= 1.75d0) then
        tmp = 0.16666666666666666d0 * ((1.0d0 + beta) / (beta + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (beta + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.75) {
		tmp = 0.16666666666666666 * ((1.0 + beta) / (beta + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.75:
		tmp = 0.16666666666666666 * ((1.0 + beta) / (beta + 2.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.75)
		tmp = Float64(0.16666666666666666 * Float64(Float64(1.0 + beta) / Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + 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 <= 1.75)
		tmp = 0.16666666666666666 * ((1.0 + beta) / (beta + 2.0));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (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, 1.75], N[(0.16666666666666666 * N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75

   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/ 99.9%

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

      2. associate-/r* 95.2%

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

      3. +-commutative 95.2%

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

      4. +-commutative 95.2%

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

      5. *-commutative 95.2%

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

      6. associate-+l+ 95.2%

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

      7. associate-+r+ 95.2%

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

      8. *-commutative 95.2%

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

      9. distribute-rgt1-in 95.2%

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

      10. +-commutative 95.2%

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

      11. *-commutative 95.2%

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

      12. distribute-rgt1-in 95.2%

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

      13. +-commutative 95.2%

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

      14. metadata-eval 95.2%

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

      15. associate-+l+ 95.2%

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

      16. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in beta around 0 94.3%

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

   5. Taylor expanded in alpha around 0 69.9%

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

   if 1.75 < beta

   1. Initial program 81.5%

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

   2. Taylor expanded in beta around inf 84.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1 + \beta}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 12: 45.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.16666666666666666 \cdot \frac{1 + \alpha}{\alpha + 2} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
}

Fortran

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

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 * Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l/ 92.8%

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

   2. associate-/r* 84.2%

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

   3. +-commutative 84.2%

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

   4. +-commutative 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-+l+ 84.2%

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

   7. associate-+r+ 84.2%

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

   8. *-commutative 84.2%

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

   9. distribute-rgt1-in 84.2%

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

   10. +-commutative 84.2%

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

   11. *-commutative 84.2%

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

   12. distribute-rgt1-in 84.2%

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

   13. +-commutative 84.2%

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

   14. metadata-eval 84.2%

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

   15. associate-+l+ 84.2%

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

   16. *-commutative 84.2%

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

3. Simplified 84.2%

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

4. Taylor expanded in beta around 0 66.3%

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

5. Taylor expanded in alpha around 0 49.2%

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

6. Taylor expanded in beta around 0 49.0%

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

7. Final simplification 49.0%

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

Alternative 13: 44.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.16666666666666666 \cdot \frac{1 + \beta}{\beta + 2} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 * ((1.0 + beta) / (beta + 2.0));
}

Fortran

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

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 * ((1.0 + beta) / (beta + 2.0))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 * Float64(Float64(1.0 + beta) / Float64(beta + 2.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l/ 92.8%

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

   2. associate-/r* 84.2%

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

   3. +-commutative 84.2%

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

   4. +-commutative 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-+l+ 84.2%

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

   7. associate-+r+ 84.2%

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

   8. *-commutative 84.2%

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

   9. distribute-rgt1-in 84.2%

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

   10. +-commutative 84.2%

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

   11. *-commutative 84.2%

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

   12. distribute-rgt1-in 84.2%

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

   13. +-commutative 84.2%

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

   14. metadata-eval 84.2%

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

   15. associate-+l+ 84.2%

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

   16. *-commutative 84.2%

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

3. Simplified 84.2%

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

4. Taylor expanded in beta around 0 66.3%

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

5. Taylor expanded in alpha around 0 49.0%

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

6. Final simplification 49.0%

   \[\leadsto 0.16666666666666666 \cdot \frac{1 + \beta}{\beta + 2} \]

Alternative 14: 46.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / ((alpha + 2.0) * (alpha + 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 = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / ((alpha + 2.0) * (alpha + 3.0));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / ((alpha + 2.0) * (alpha + 3.0))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
end

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l/ 92.8%

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

   2. *-commutative 92.8%

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

   3. +-commutative 92.8%

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

   4. +-commutative 92.8%

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

   5. +-commutative 92.8%

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

   6. +-commutative 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in alpha around 0 86.0%

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

   1. +-commutative 86.0%

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

6. Simplified 86.0%

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

7. Taylor expanded in beta around 0 63.8%

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

8. Final simplification 63.8%

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

Alternative 15: 10.6% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l/ 92.8%

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

   2. associate-/r* 84.2%

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

   3. +-commutative 84.2%

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

   4. +-commutative 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-+l+ 84.2%

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

   7. associate-+r+ 84.2%

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

   8. *-commutative 84.2%

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

   9. distribute-rgt1-in 84.2%

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

   10. +-commutative 84.2%

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

   11. *-commutative 84.2%

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

   12. distribute-rgt1-in 84.2%

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

   13. +-commutative 84.2%

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

   14. metadata-eval 84.2%

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

   15. associate-+l+ 84.2%

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

   16. *-commutative 84.2%

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

3. Simplified 84.2%

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

4. Taylor expanded in beta around 0 66.3%

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

5. Taylor expanded in alpha around 0 49.2%

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

6. Taylor expanded in beta around inf 10.8%

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

7. Taylor expanded in alpha around 0 11.2%

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

8. Final simplification 11.2%

   \[\leadsto 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023305 
(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)))