Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.6%

Time: 15.0s

Alternatives: 19

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) + \frac{-1 - \alpha}{\frac{\beta}{\alpha + 2}}\right)}{t_0}}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1.4e+23)
     (*
      (/ (+ beta 1.0) (+ alpha (+ beta 3.0)))
      (/
       (+ 1.0 alpha)
       (+ (* beta beta) (* (+ alpha 2.0) (+ (+ alpha 2.0) (* beta 2.0))))))
     (/
      (/
       (+
        (/ 1.0 beta)
        (+
         (+ (+ 1.0 alpha) (/ alpha beta))
         (/ (- -1.0 alpha) (/ beta (+ alpha 2.0)))))
       t_0)
      (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.4e+23) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + 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 = 2.0d0 + (beta + alpha)
    if (beta <= 1.4d+23) then
        tmp = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * ((1.0d0 + alpha) / ((beta * beta) + ((alpha + 2.0d0) * ((alpha + 2.0d0) + (beta * 2.0d0)))))
    else
        tmp = (((1.0d0 / beta) + (((1.0d0 + alpha) + (alpha / beta)) + (((-1.0d0) - alpha) / (beta / (alpha + 2.0d0))))) / t_0) / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.4e+23) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 1.4e+23:
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))))
	else:
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.4e+23)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 + alpha) / Float64(Float64(beta * beta) + Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 2.0) + Float64(beta * 2.0))))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / beta) + Float64(Float64(Float64(1.0 + alpha) + Float64(alpha / beta)) + Float64(Float64(-1.0 - alpha) / Float64(beta / Float64(alpha + 2.0))))) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 1.4e+23)
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	else
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.4e+23], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(beta * beta), $MachinePrecision] + N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(N[(N[(1.0 + alpha), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] / N[(beta / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.4e23

   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/ 99.5%

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

      2. associate-/l/ 90.9%

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

      3. associate-+l+ 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-+r+ 90.9%

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

      6. associate-+l+ 90.9%

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

      7. distribute-rgt1-in 90.9%

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

      8. *-rgt-identity 90.9%

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

      9. distribute-lft-out 90.9%

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

      10. +-commutative 90.9%

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

      11. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around -inf 99.4%

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

      1. unpow2 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. associate-*r* 99.4%

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

      5. distribute-rgt-out 99.4%

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

      6. +-commutative 99.4%

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

      7. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   if 1.4e23 < beta

   1. Initial program 84.8%

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

   2. Taylor expanded in beta around inf 86.0%

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

      1. associate--l+ 86.0%

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

      2. +-commutative 86.0%

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

      3. associate-/l* 90.7%

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

      4. +-commutative 90.7%

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

   4. Simplified 90.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) + \frac{-1 - \alpha}{\frac{\beta}{\alpha + 2}}\right)}{2 + \left(\beta + \alpha\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) + \left(\alpha + 2\right) \cdot \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

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 3.0))))
   (if (<= beta 4e+153)
     (*
      (/ (+ beta 1.0) t_0)
      (/
       (+ 1.0 alpha)
       (+ (* beta beta) (* (+ alpha 2.0) (+ (+ alpha 2.0) (* beta 2.0))))))
     (*
      (/
       (+
        (/ 1.0 beta)
        (+
         (+ (+ 1.0 alpha) (/ alpha beta))
         (* (+ alpha 2.0) (/ (- -1.0 alpha) beta))))
       (+ alpha (+ beta 2.0)))
      (/ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 4e+153) {
		tmp = ((beta + 1.0) / t_0) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((alpha + 2.0) * ((-1.0 - alpha) / beta)))) / (alpha + (beta + 2.0))) * (1.0 / 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 + 3.0d0)
    if (beta <= 4d+153) then
        tmp = ((beta + 1.0d0) / t_0) * ((1.0d0 + alpha) / ((beta * beta) + ((alpha + 2.0d0) * ((alpha + 2.0d0) + (beta * 2.0d0)))))
    else
        tmp = (((1.0d0 / beta) + (((1.0d0 + alpha) + (alpha / beta)) + ((alpha + 2.0d0) * (((-1.0d0) - alpha) / beta)))) / (alpha + (beta + 2.0d0))) * (1.0d0 / 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 + 3.0);
	double tmp;
	if (beta <= 4e+153) {
		tmp = ((beta + 1.0) / t_0) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((alpha + 2.0) * ((-1.0 - alpha) / beta)))) / (alpha + (beta + 2.0))) * (1.0 / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 4e+153:
		tmp = ((beta + 1.0) / t_0) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))))
	else:
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((alpha + 2.0) * ((-1.0 - alpha) / beta)))) / (alpha + (beta + 2.0))) * (1.0 / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 4e+153)
		tmp = Float64(Float64(Float64(beta + 1.0) / t_0) * Float64(Float64(1.0 + alpha) / Float64(Float64(beta * beta) + Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 2.0) + Float64(beta * 2.0))))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / beta) + Float64(Float64(Float64(1.0 + alpha) + Float64(alpha / beta)) + Float64(Float64(alpha + 2.0) * Float64(Float64(-1.0 - alpha) / beta)))) / Float64(alpha + Float64(beta + 2.0))) * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 4e+153)
		tmp = ((beta + 1.0) / t_0) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	else
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((alpha + 2.0) * ((-1.0 - alpha) / beta)))) / (alpha + (beta + 2.0))) * (1.0 / 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 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4e+153], N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(beta * beta), $MachinePrecision] + N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(N[(N[(1.0 + alpha), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e153

   1. Initial program 98.8%

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

      1. associate-/l/ 98.5%

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

      2. associate-/l/ 84.2%

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

      3. associate-+l+ 84.2%

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

      4. +-commutative 84.2%

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

      5. associate-+r+ 84.2%

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

      6. associate-+l+ 84.2%

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

      7. distribute-rgt1-in 84.2%

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

      8. *-rgt-identity 84.2%

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

      9. distribute-lft-out 84.2%

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

      10. +-commutative 84.2%

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

      11. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around -inf 99.2%

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

      1. unpow2 99.2%

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

      2. +-commutative 99.2%

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

      3. unpow2 99.2%

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

      4. associate-*r* 99.2%

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

      5. distribute-rgt-out 99.2%

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

      6. +-commutative 99.2%

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

      7. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   if 4e153 < beta

   1. Initial program 80.6%

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

   2. Taylor expanded in beta around inf 87.7%

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

      1. associate--l+ 87.7%

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

      2. +-commutative 87.7%

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

      3. associate-/l* 94.7%

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

      4. +-commutative 94.7%

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

   4. Simplified 94.7%

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

      1. div-inv 94.7%

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

      2. +-commutative 94.7%

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

      3. associate-/r/ 94.7%

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

      4. +-commutative 94.7%

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

      5. metadata-eval 94.7%

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

      6. associate-+r+ 94.7%

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

      7. metadata-eval 94.7%

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

      8. associate-+l+ 94.7%

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

      9. metadata-eval 94.7%

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

      10. associate-+r+ 94.7%

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

   6. Applied egg-rr 94.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) + \left(\alpha + 2\right) \cdot \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55e+23)
   (*
    (/ (+ beta 1.0) (+ alpha (+ beta 3.0)))
    (/
     (+ 1.0 alpha)
     (+ (* beta beta) (* (+ alpha 2.0) (+ (+ alpha 2.0) (* beta 2.0))))))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55e+23) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1.55d+23) then
        tmp = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * ((1.0d0 + alpha) / ((beta * beta) + ((alpha + 2.0d0) * ((alpha + 2.0d0) + (beta * 2.0d0)))))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (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 <= 1.55e+23) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55e+23:
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55e+23)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 + alpha) / Float64(Float64(beta * beta) + Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 2.0) + Float64(beta * 2.0))))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 <= 1.55e+23)
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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, 1.55e+23], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(beta * beta), $MachinePrecision] + N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.54999999999999985e23

   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/ 99.5%

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

      2. associate-/l/ 90.9%

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

      3. associate-+l+ 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-+r+ 90.9%

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

      6. associate-+l+ 90.9%

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

      7. distribute-rgt1-in 90.9%

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

      8. *-rgt-identity 90.9%

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

      9. distribute-lft-out 90.9%

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

      10. +-commutative 90.9%

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

      11. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around -inf 99.4%

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

      1. unpow2 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. associate-*r* 99.4%

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

      5. distribute-rgt-out 99.4%

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

      6. +-commutative 99.4%

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

      7. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   if 1.54999999999999985e23 < beta

   1. Initial program 84.8%

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

   2. Taylor expanded in beta around inf 90.7%

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

   3. Taylor expanded in alpha around 0 90.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+17}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1 + \alpha}{t_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

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 1.8e+17)
     (* (+ beta 1.0) (/ (/ (+ 1.0 alpha) t_0) (* (+ alpha (+ beta 3.0)) t_0)))
     (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ 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 <= 1.8e+17) {
		tmp = (beta + 1.0) * (((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1.8d+17) then
        tmp = (beta + 1.0d0) * (((1.0d0 + alpha) / t_0) / ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (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 <= 1.8e+17) {
		tmp = (beta + 1.0) * (((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1.8e+17:
		tmp = (beta + 1.0) * (((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1.8e+17)
		tmp = Float64(Float64(beta + 1.0) * Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 <= 1.8e+17)
		tmp = (beta + 1.0) * (((1.0 + alpha) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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, 1.8e+17], N[(N[(beta + 1.0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8e17

   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/ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. associate-+l+ 99.5%

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

      6. distribute-rgt1-in 99.5%

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

      7. *-rgt-identity 99.5%

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

      8. distribute-lft-out 99.5%

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

      9. +-commutative 99.5%

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

      10. associate-*r/ 99.5%

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

      11. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   if 1.8e17 < beta

   1. Initial program 85.3%

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

   2. Taylor expanded in beta around inf 89.9%

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

   3. Taylor expanded in alpha around 0 89.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+17}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

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 1.55e+23)
     (* (/ (+ beta 1.0) (+ alpha (+ beta 3.0))) (/ (+ 1.0 alpha) (* t_0 t_0)))
     (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ 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 <= 1.55e+23) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / (t_0 * t_0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1.55d+23) then
        tmp = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * ((1.0d0 + alpha) / (t_0 * t_0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (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 <= 1.55e+23) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / (t_0 * t_0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1.55e+23:
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / (t_0 * t_0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1.55e+23)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 + alpha) / Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 <= 1.55e+23)
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / (t_0 * t_0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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, 1.55e+23], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.54999999999999985e23

   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/ 99.5%

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

      2. associate-/l/ 90.9%

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

      3. associate-+l+ 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-+r+ 90.9%

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

      6. associate-+l+ 90.9%

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

      7. distribute-rgt1-in 90.9%

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

      8. *-rgt-identity 90.9%

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

      9. distribute-lft-out 90.9%

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

      10. +-commutative 90.9%

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

      11. times-frac 99.4%

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

   3. Simplified 99.4%

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

   if 1.54999999999999985e23 < beta

   1. Initial program 84.8%

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

   2. Taylor expanded in beta around inf 90.7%

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

   3. Taylor expanded in alpha around 0 90.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\left(\left(\beta + 1\right) \cdot \left(1 + \alpha\right)\right) \cdot \frac{\frac{\frac{1}{\beta + 3}}{\beta + 2}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4e+42)
   (*
    (* (+ beta 1.0) (+ 1.0 alpha))
    (/ (/ (/ 1.0 (+ beta 3.0)) (+ beta 2.0)) (+ alpha (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+42) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) * (((1.0 / (beta + 3.0)) / (beta + 2.0)) / (alpha + (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 4d+42) then
        tmp = ((beta + 1.0d0) * (1.0d0 + alpha)) * (((1.0d0 / (beta + 3.0d0)) / (beta + 2.0d0)) / (alpha + (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (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 <= 4e+42) {
		tmp = ((beta + 1.0) * (1.0 + alpha)) * (((1.0 / (beta + 3.0)) / (beta + 2.0)) / (alpha + (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4e+42:
		tmp = ((beta + 1.0) * (1.0 + alpha)) * (((1.0 / (beta + 3.0)) / (beta + 2.0)) / (alpha + (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4e+42)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(1.0 + alpha)) * Float64(Float64(Float64(1.0 / Float64(beta + 3.0)) / Float64(beta + 2.0)) / Float64(alpha + Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 <= 4e+42)
		tmp = ((beta + 1.0) * (1.0 + alpha)) * (((1.0 / (beta + 3.0)) / (beta + 2.0)) / (alpha + (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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, 4e+42], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.00000000000000018e42

   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/ 99.5%

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

      2. associate-/r* 91.1%

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

      3. associate-+l+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

      6. associate-+l+ 91.1%

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

      7. distribute-rgt1-in 91.1%

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

      8. *-rgt-identity 91.1%

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

      9. distribute-lft-out 91.1%

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

      10. *-commutative 91.1%

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

      11. metadata-eval 91.1%

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

      12. associate-+l+ 91.1%

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

      13. +-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in alpha around 0 60.9%

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

      1. div-inv 60.9%

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

      2. +-commutative 60.9%

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

      3. associate-+r+ 60.9%

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

      4. metadata-eval 60.9%

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

      5. *-commutative 60.9%

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

      6. metadata-eval 60.9%

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

      7. associate-+r+ 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. +-commutative 60.9%

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

      2. +-commutative 60.9%

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

      3. *-commutative 60.9%

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

      4. +-commutative 60.9%

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

      5. associate-/r* 60.4%

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

      6. associate-/r* 60.4%

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

      7. +-commutative 60.4%

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

      8. +-commutative 60.4%

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

   8. Simplified 60.4%

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

   if 4.00000000000000018e42 < beta

   1. Initial program 84.1%

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

   2. Taylor expanded in beta around inf 90.2%

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

   3. Taylor expanded in alpha around 0 90.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\left(\left(\beta + 1\right) \cdot \left(1 + \alpha\right)\right) \cdot \frac{\frac{\frac{1}{\beta + 3}}{\beta + 2}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+23}:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot t_0} \cdot \frac{\beta + 1}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

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 1e+23)
     (* (/ (+ 1.0 alpha) (* t_0 t_0)) (/ (+ beta 1.0) (+ beta 3.0)))
     (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ 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 <= 1e+23) {
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1d+23) then
        tmp = ((1.0d0 + alpha) / (t_0 * t_0)) * ((beta + 1.0d0) / (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (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 <= 1e+23) {
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1e+23:
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 1e+23)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(t_0 * t_0)) * Float64(Float64(beta + 1.0) / Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 <= 1e+23)
		tmp = ((1.0 + alpha) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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, 1e+23], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999992e22

   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/ 99.5%

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

      2. associate-/l/ 90.9%

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

      3. associate-+l+ 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-+r+ 90.9%

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

      6. associate-+l+ 90.9%

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

      7. distribute-rgt1-in 90.9%

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

      8. *-rgt-identity 90.9%

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

      9. distribute-lft-out 90.9%

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

      10. +-commutative 90.9%

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

      11. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in alpha around 0 81.0%

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

   if 9.9999999999999992e22 < beta

   1. Initial program 84.8%

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

   2. Taylor expanded in beta around inf 90.7%

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

   3. Taylor expanded in alpha around 0 90.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+23}:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \cdot \frac{\beta + 1}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 8: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.9 \cdot 10^{+15}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.9e+15)
   (*
    (+ beta 1.0)
    (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.9e+15) {
		tmp = (beta + 1.0) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 7.9d+15) then
        tmp = (beta + 1.0d0) * (((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (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 <= 7.9e+15) {
		tmp = (beta + 1.0) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.9e+15:
		tmp = (beta + 1.0) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.9e+15)
		tmp = Float64(Float64(beta + 1.0) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 <= 7.9e+15)
		tmp = (beta + 1.0) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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, 7.9e+15], N[(N[(beta + 1.0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.9e15

   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/ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. associate-+l+ 99.5%

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

      6. distribute-rgt1-in 99.5%

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

      7. *-rgt-identity 99.5%

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

      8. distribute-lft-out 99.5%

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

      9. +-commutative 99.5%

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

      10. associate-*r/ 99.5%

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

      11. associate-*r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in alpha around 0 59.3%

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

   if 7.9e15 < beta

   1. Initial program 85.3%

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

   2. Taylor expanded in beta around inf 89.9%

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

   3. Taylor expanded in alpha around 0 89.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.9 \cdot 10^{+15}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 9: 97.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{1 + \alpha}{\left(4 + \alpha \cdot 4\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ (+ 1.0 alpha) (* (+ 4.0 (* alpha 4.0)) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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 <= 5.2d0) then
        tmp = (1.0d0 + alpha) / ((4.0d0 + (alpha * 4.0d0)) * (alpha + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (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 <= 5.2) {
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(4.0 + Float64(alpha * 4.0)) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / 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 <= 5.2)
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (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, 5.2], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(4.0 + N[(alpha * 4.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. associate-/l/ 91.1%

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

      3. associate-+l+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

      6. associate-+l+ 91.1%

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

      7. distribute-rgt1-in 91.1%

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

      8. *-rgt-identity 91.1%

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

      9. distribute-lft-out 91.1%

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

      10. +-commutative 91.1%

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

      11. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around -inf 99.4%

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

      1. unpow2 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. associate-*r* 99.4%

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

      5. distribute-rgt-out 99.4%

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

      6. +-commutative 99.4%

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

      7. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in alpha around 0 61.8%

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

   8. Taylor expanded in beta around 0 79.4%

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

      1. +-commutative 79.4%

         \[\leadsto \frac{1 + \alpha}{\left(4 + 4 \cdot \alpha\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   10. Simplified 79.4%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(4 + 4 \cdot \alpha\right) \cdot \left(\alpha + 3\right)}} \]

   if 5.20000000000000018 < beta

   1. Initial program 85.8%

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

   2. Taylor expanded in beta around inf 87.4%

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

   3. Taylor expanded in alpha around 0 87.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{1 + \alpha}{\left(4 + \alpha \cdot 4\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 10: 96.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{1 + \alpha}{\left(4 + \alpha \cdot 4\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ (+ 1.0 alpha) (* (+ 4.0 (* alpha 4.0)) (+ alpha 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0));
	} else {
		tmp = ((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 <= 6.2d0) then
        tmp = (1.0d0 + alpha) / ((4.0d0 + (alpha * 4.0d0)) * (alpha + 3.0d0))
    else
        tmp = ((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 <= 6.2) {
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0))
	else:
		tmp = ((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 <= 6.2)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(4.0 + Float64(alpha * 4.0)) * Float64(alpha + 3.0)));
	else
		tmp = 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 <= 6.2)
		tmp = (1.0 + alpha) / ((4.0 + (alpha * 4.0)) * (alpha + 3.0));
	else
		tmp = ((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, 6.2], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(4.0 + N[(alpha * 4.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   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/ 99.5%

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

      2. associate-/l/ 91.1%

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

      3. associate-+l+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

      6. associate-+l+ 91.1%

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

      7. distribute-rgt1-in 91.1%

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

      8. *-rgt-identity 91.1%

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

      9. distribute-lft-out 91.1%

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

      10. +-commutative 91.1%

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

      11. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in beta around -inf 99.4%

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

      1. unpow2 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. associate-*r* 99.4%

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

      5. distribute-rgt-out 99.4%

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

      6. +-commutative 99.4%

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

      7. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in alpha around 0 61.8%

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

   8. Taylor expanded in beta around 0 79.4%

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

      1. +-commutative 79.4%

         \[\leadsto \frac{1 + \alpha}{\left(4 + 4 \cdot \alpha\right) \cdot \color{blue}{\left(\alpha + 3\right)}} \]

   10. Simplified 79.4%

       \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(4 + 4 \cdot \alpha\right) \cdot \left(\alpha + 3\right)}} \]

   if 6.20000000000000018 < beta

   1. Initial program 85.8%

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

   2. Taylor expanded in beta around inf 82.9%

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

      1. associate--l+ 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-/l* 87.3%

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

      4. +-commutative 87.3%

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

   4. Simplified 87.3%

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

   5. Taylor expanded in beta around inf 87.0%

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

      1. expm1-log1p-u 87.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 58.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. associate-/l/ 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \beta}}\right)} - 1 \]

      4. +-commutative 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\alpha + 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \beta}\right)} - 1 \]

      5. metadata-eval 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \beta}\right)} - 1 \]

      6. associate-+l+ 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \beta}\right)} - 1 \]

      7. metadata-eval 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \beta}\right)} - 1 \]

      8. associate-+r+ 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \beta}\right)} - 1 \]

   7. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \beta}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 84.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \beta}\right)\right)} \]

      2. expm1-log1p 84.9%

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

      3. +-commutative 84.9%

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

      4. *-lft-identity 84.9%

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

      5. times-frac 86.9%

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

      6. +-commutative 86.9%

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

      7. *-commutative 86.9%

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

      8. associate-*r/ 87.0%

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

      9. *-rgt-identity 87.0%

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

      10. +-commutative 87.0%

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

      11. +-commutative 87.0%

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

      12. associate-+l+ 87.0%

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

      13. +-commutative 87.0%

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

   9. Simplified 87.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{1 + \alpha}{\left(4 + \alpha \cdot 4\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 11: 96.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	} else {
		tmp = ((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 <= 2.3d0) then
        tmp = 0.16666666666666666d0 * ((1.0d0 + alpha) / (alpha + 2.0d0))
    else
        tmp = ((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 <= 2.3) {
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0))
	else:
		tmp = ((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 <= 2.3)
		tmp = Float64(0.16666666666666666 * Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)));
	else
		tmp = 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 <= 2.3)
		tmp = 0.16666666666666666 * ((1.0 + alpha) / (alpha + 2.0));
	else
		tmp = ((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, 2.3], N[(0.16666666666666666 * N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   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/ 99.4%

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

      2. associate-/r* 91.1%

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

      3. associate-+l+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

      6. associate-+l+ 91.1%

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

      7. distribute-rgt1-in 91.1%

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

      8. *-rgt-identity 91.1%

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

      9. distribute-lft-out 91.1%

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

      10. *-commutative 91.1%

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

      11. metadata-eval 91.1%

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

      12. associate-+l+ 91.1%

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

      13. +-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in alpha around 0 60.3%

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

   5. Taylor expanded in beta around 0 57.9%

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

   if 2.2999999999999998 < beta

   1. Initial program 85.8%

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

   2. Taylor expanded in beta around inf 82.9%

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

      1. associate--l+ 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-/l* 87.3%

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

      4. +-commutative 87.3%

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

   4. Simplified 87.3%

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

   5. Taylor expanded in beta around inf 87.0%

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

      1. expm1-log1p-u 87.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)\right)} \]

      2. expm1-udef 58.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right)} - 1} \]

      3. associate-/l/ 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \beta}}\right)} - 1 \]

      4. +-commutative 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\alpha + 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \beta}\right)} - 1 \]

      5. metadata-eval 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \beta}\right)} - 1 \]

      6. associate-+l+ 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \beta}\right)} - 1 \]

      7. metadata-eval 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \beta}\right)} - 1 \]

      8. associate-+r+ 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \beta}\right)} - 1 \]

   7. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \beta}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 84.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \beta}\right)\right)} \]

      2. expm1-log1p 84.9%

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

      3. +-commutative 84.9%

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

      4. *-lft-identity 84.9%

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

      5. times-frac 86.9%

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

      6. +-commutative 86.9%

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

      7. *-commutative 86.9%

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

      8. associate-*r/ 87.0%

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

      9. *-rgt-identity 87.0%

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

      10. +-commutative 87.0%

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

      11. +-commutative 87.0%

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

      12. associate-+l+ 87.0%

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

      13. +-commutative 87.0%

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

   9. Simplified 87.0%

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

4. Final simplification 68.3%

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

Alternative 12: 96.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   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/ 99.4%

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

      2. associate-/r* 91.1%

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

      3. associate-+l+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+r+ 91.1%

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

      6. associate-+l+ 91.1%

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

      7. distribute-rgt1-in 91.1%

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

      8. *-rgt-identity 91.1%

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

      9. distribute-lft-out 91.1%

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

      10. *-commutative 91.1%

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

      11. metadata-eval 91.1%

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

      12. associate-+l+ 91.1%

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

      13. +-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in alpha around 0 60.3%

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

   5. Taylor expanded in beta around 0 57.9%

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

   if 3.60000000000000009 < beta

   1. Initial program 85.8%

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

   2. Taylor expanded in beta around inf 82.9%

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

      1. associate--l+ 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-/l* 87.3%

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

      4. +-commutative 87.3%

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

   4. Simplified 87.3%

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

   5. Taylor expanded in beta around inf 87.0%

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

   6. Taylor expanded in beta around inf 86.9%

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

4. Final simplification 68.2%

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

Alternative 13: 59.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.2], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   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. Taylor expanded in beta around inf 14.0%

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

   3. Taylor expanded in alpha around 0 12.7%

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

      1. associate-/r* 12.7%

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

      2. +-commutative 12.7%

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

   5. Simplified 12.7%

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

   6. Taylor expanded in beta around 0 12.8%

      \[\leadsto \color{blue}{-0.1388888888888889 \cdot \beta + 0.16666666666666666} \]

   if 1.19999999999999996 < beta

   1. Initial program 85.8%

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

      1. associate-/l/ 83.4%

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

      2. associate-+l+ 83.4%

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

      3. +-commutative 83.4%

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

      4. associate-+r+ 83.4%

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

      5. associate-+l+ 83.4%

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

      6. distribute-rgt1-in 83.4%

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

      7. *-rgt-identity 83.4%

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

      8. distribute-lft-out 83.4%

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

      9. +-commutative 83.4%

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

      10. associate-*r/ 92.0%

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

      11. associate-*r/ 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in beta around inf 83.2%

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

      1. unpow2 83.2%

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

   6. Simplified 83.2%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14: 62.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.2], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   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. Taylor expanded in beta around inf 14.0%

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

   3. Taylor expanded in alpha around 0 12.7%

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

      1. associate-/r* 12.7%

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

      2. +-commutative 12.7%

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

   5. Simplified 12.7%

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

   6. Taylor expanded in beta around 0 12.8%

      \[\leadsto \color{blue}{-0.1388888888888889 \cdot \beta + 0.16666666666666666} \]

   if 1.19999999999999996 < beta

   1. Initial program 85.8%

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

   2. Taylor expanded in beta around inf 82.9%

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

      1. associate--l+ 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-/l* 87.3%

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

      4. +-commutative 87.3%

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

   4. Simplified 87.3%

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

   5. Taylor expanded in beta around inf 87.0%

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

   6. Taylor expanded in beta around inf 86.9%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 15: 12.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.15)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ 0.3333333333333333 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 0.3333333333333333 / 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 <= 1.15d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = 0.3333333333333333d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.15:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = 0.3333333333333333 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.15)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(0.3333333333333333 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.15)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = 0.3333333333333333 / 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, 1.15], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1499999999999999

   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. Taylor expanded in beta around inf 14.0%

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

   3. Taylor expanded in alpha around 0 12.7%

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

      1. associate-/r* 12.7%

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

      2. +-commutative 12.7%

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

   5. Simplified 12.7%

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

   6. Taylor expanded in beta around 0 12.8%

      \[\leadsto \color{blue}{-0.1388888888888889 \cdot \beta + 0.16666666666666666} \]

   if 1.1499999999999999 < beta

   1. Initial program 85.8%

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

   2. Taylor expanded in beta around inf 82.9%

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

      1. associate--l+ 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-/l* 87.3%

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

      4. +-commutative 87.3%

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

   4. Simplified 87.3%

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

   5. Taylor expanded in beta around inf 87.0%

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

   6. Taylor expanded in alpha around 0 80.4%

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

   7. Taylor expanded in beta around 0 7.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \]

Alternative 16: 56.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.2], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   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. Taylor expanded in beta around inf 14.0%

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

   3. Taylor expanded in alpha around 0 12.7%

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

      1. associate-/r* 12.7%

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

      2. +-commutative 12.7%

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

   5. Simplified 12.7%

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

   6. Taylor expanded in beta around 0 12.8%

      \[\leadsto \color{blue}{-0.1388888888888889 \cdot \beta + 0.16666666666666666} \]

   if 1.19999999999999996 < beta

   1. Initial program 85.8%

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

      1. associate-/l/ 83.4%

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

      2. associate-+l+ 83.4%

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

      3. +-commutative 83.4%

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

      4. associate-+r+ 83.4%

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

      5. associate-+l+ 83.4%

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

      6. distribute-rgt1-in 83.4%

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

      7. *-rgt-identity 83.4%

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

      8. distribute-lft-out 83.4%

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

      9. +-commutative 83.4%

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

      10. associate-*r/ 92.0%

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

      11. associate-*r/ 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in beta around inf 83.2%

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

      1. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in alpha around 0 80.3%

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

      1. unpow2 80.3%

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

   9. Simplified 80.3%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 17: 56.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (+ 0.16666666666666666 (* beta -0.1388888888888889))
   (/ (/ 1.0 beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d0) then
        tmp = 0.16666666666666666d0 + (beta * (-0.1388888888888889d0))
    else
        tmp = (1.0d0 / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889)
	else:
		tmp = (1.0 / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.16666666666666666 + Float64(beta * -0.1388888888888889));
	else
		tmp = Float64(Float64(1.0 / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.16666666666666666 + (beta * -0.1388888888888889);
	else
		tmp = (1.0 / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.2], N[(0.16666666666666666 + N[(beta * -0.1388888888888889), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   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. Taylor expanded in beta around inf 14.0%

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

   3. Taylor expanded in alpha around 0 12.7%

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

      1. associate-/r* 12.7%

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

      2. +-commutative 12.7%

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

   5. Simplified 12.7%

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

   6. Taylor expanded in beta around 0 12.8%

      \[\leadsto \color{blue}{-0.1388888888888889 \cdot \beta + 0.16666666666666666} \]

   if 1.19999999999999996 < beta

   1. Initial program 85.8%

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

      1. associate-/l/ 83.4%

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

      2. associate-+l+ 83.4%

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

      3. +-commutative 83.4%

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

      4. associate-+r+ 83.4%

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

      5. associate-+l+ 83.4%

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

      6. distribute-rgt1-in 83.4%

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

      7. *-rgt-identity 83.4%

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

      8. distribute-lft-out 83.4%

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

      9. +-commutative 83.4%

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

      10. associate-*r/ 92.0%

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

      11. associate-*r/ 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in beta around inf 83.2%

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

      1. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in alpha around 0 80.3%

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

      1. unpow2 80.3%

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

      2. associate-/r* 81.5%

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

   9. Simplified 81.5%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 18: 12.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \end{array} \]

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 (/ 0.3333333333333333 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 0.3333333333333333 / 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
    else
        tmp = 0.3333333333333333d0 / 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;
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.16666666666666666
	else:
		tmp = 0.3333333333333333 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.16666666666666666;
	else
		tmp = Float64(0.3333333333333333 / 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;
	else
		tmp = 0.3333333333333333 / 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], 0.16666666666666666, N[(0.3333333333333333 / beta), $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. Taylor expanded in beta around inf 14.0%

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

   3. Taylor expanded in alpha around 0 12.7%

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

      1. associate-/r* 12.7%

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

      2. +-commutative 12.7%

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

   5. Simplified 12.7%

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

   6. Taylor expanded in beta around 0 12.7%

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

   if 2 < beta

   1. Initial program 85.8%

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

   2. Taylor expanded in beta around inf 82.9%

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

      1. associate--l+ 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-/l* 87.3%

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

      4. +-commutative 87.3%

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

   4. Simplified 87.3%

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

   5. Taylor expanded in beta around inf 87.0%

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

   6. Taylor expanded in alpha around 0 80.4%

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

   7. Taylor expanded in beta around 0 7.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \]

Alternative 19: 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 94.8%

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

2. Taylor expanded in beta around inf 40.0%

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

3. Taylor expanded in alpha around 0 36.8%

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

   1. associate-/r* 37.2%

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

   2. +-commutative 37.2%

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

5. Simplified 37.2%

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

6. Taylor expanded in beta around 0 9.6%

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

7. Final simplification 9.6%

   \[\leadsto 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023196 
(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)))