Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.7%

Time: 16.5s

Alternatives: 19

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \beta}{t\_0} \cdot \frac{\alpha + 1}{t\_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 5.8e+152)
     (* (/ (+ 1.0 beta) t_0) (/ (+ alpha 1.0) (* t_0 (+ (+ alpha beta) 3.0))))
     (* (/ (+ alpha 1.0) t_0) (/ (- 1.0 (* 2.0 (/ alpha beta))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 5.8e+152) {
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * ((alpha + beta) + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 5.8d+152) then
        tmp = ((1.0d0 + beta) / t_0) * ((alpha + 1.0d0) / (t_0 * ((alpha + beta) + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 5.8e+152) {
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * ((alpha + beta) + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 5.8e+152:
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * ((alpha + beta) + 3.0)))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 5.8e+152)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(Float64(alpha + 1.0) / Float64(t_0 * Float64(Float64(alpha + beta) + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 5.8e+152)
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * ((alpha + beta) + 3.0)));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (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_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.8e+152], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.7999999999999997e152

   1. Initial program 98.0%

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

      1. associate-/l/ 97.1%

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

      2. +-commutative 97.1%

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

      3. +-commutative 97.1%

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

      4. associate-+r+ 97.1%

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

      5. associate-+r+ 97.1%

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

      6. associate-+r+ 97.1%

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

      7. distribute-rgt1-in 97.1%

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

      8. +-commutative 97.1%

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

      9. *-commutative 97.1%

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

      10. distribute-rgt1-in 97.1%

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

      11. +-commutative 97.1%

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

      12. metadata-eval 97.1%

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

      13. associate-+l+ 97.1%

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

   3. Simplified 97.1%

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

      1. associate-+r+ 97.2%

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

      2. metadata-eval 97.2%

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

      3. associate-/l/ 85.1%

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

      4. times-frac 99.0%

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

      5. associate-+r+ 99.0%

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

      6. +-commutative 99.0%

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

      7. associate-+r+ 99.0%

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

      8. +-commutative 99.0%

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

      9. metadata-eval 99.0%

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

      10. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   if 5.7999999999999997e152 < beta

   1. Initial program 71.1%

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

      1. associate-/l/ 64.4%

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

      2. +-commutative 64.4%

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

      3. +-commutative 64.4%

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

      4. associate-+r+ 64.4%

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

      5. associate-+r+ 64.4%

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

      6. associate-+r+ 64.4%

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

      7. distribute-rgt1-in 64.4%

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

      8. +-commutative 64.4%

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

      9. *-commutative 64.4%

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

      10. distribute-rgt1-in 64.4%

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

      11. +-commutative 64.4%

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

      12. metadata-eval 64.4%

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

      13. associate-+l+ 64.4%

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

   3. Simplified 64.4%

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

      1. associate-+r+ 64.4%

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

      2. metadata-eval 64.4%

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

      3. associate-/l* 76.7%

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

      4. associate-+r+ 76.7%

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

      5. metadata-eval 76.7%

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

      6. associate-+r+ 76.7%

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

      7. metadata-eval 76.7%

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

      8. associate-+l+ 76.7%

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

      9. metadata-eval 76.7%

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

      10. times-frac 99.7%

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

      11. metadata-eval 99.7%

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

      12. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. metadata-eval 95.2%

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

      3. distribute-lft-in 95.2%

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

   9. Simplified 95.2%

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

   10. Taylor expanded in alpha around inf 95.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \beta}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+40}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 + \beta}{t\_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 1.55e+40)
     (* (/ (+ alpha 1.0) t_0) (/ (+ 1.0 beta) (* t_0 (+ (+ alpha beta) 3.0))))
     (/ (/ (+ alpha 1.0) (+ alpha (+ 2.0 beta))) (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.55e+40) {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * ((alpha + beta) + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 1.55d+40) then
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 + beta) / (t_0 * ((alpha + beta) + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (2.0d0 + beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.55e+40) {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * ((alpha + beta) + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 1.55e+40:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * ((alpha + beta) + 3.0)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1.55e+40)
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(alpha + beta) + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(2.0 + beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 1.55e+40)
		tmp = ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * ((alpha + beta) + 3.0)));
	else
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.55e+40], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5499999999999999e40

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. associate-+r+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. distribute-rgt1-in 99.9%

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

      8. +-commutative 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

      1. associate-+r+ 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/l/ 92.3%

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

      4. *-commutative 92.3%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 1.5499999999999999e40 < beta

   1. Initial program 78.1%

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

      1. associate-/l/ 72.0%

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

      2. +-commutative 72.0%

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

      3. +-commutative 72.0%

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

      4. associate-+r+ 72.0%

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

      5. associate-+r+ 72.0%

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

      6. associate-+r+ 72.0%

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

      7. distribute-rgt1-in 72.0%

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

      8. +-commutative 72.0%

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

      9. *-commutative 72.0%

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

      10. distribute-rgt1-in 72.0%

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

      11. +-commutative 72.0%

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

      12. metadata-eval 72.0%

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

      13. associate-+l+ 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in beta around inf 83.1%

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

      1. associate-+r+ 83.1%

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

      2. associate-/r* 88.4%

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

      3. +-commutative 88.4%

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

      4. +-commutative 88.4%

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

      5. associate-+r+ 88.4%

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

   7. Applied egg-rr 88.4%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+40}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 + \beta}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4e+15)
   (/
    (/ (+ 1.0 beta) (+ 2.0 beta))
    (* (+ alpha (+ 2.0 beta)) (+ alpha (+ beta 3.0))))
   (*
    (/ (+ alpha 1.0) (+ 2.0 (+ alpha beta)))
    (/ (- 1.0 (/ (* 2.0 (+ alpha 2.0)) beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4e+15) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.4d+15) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((alpha + (2.0d0 + beta)) * (alpha + (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (2.0d0 + (alpha + beta))) * ((1.0d0 - ((2.0d0 * (alpha + 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4e+15) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.4e+15:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)))
	else:
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.4e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(alpha + beta))) * Float64(Float64(1.0 - Float64(Float64(2.0 * Float64(alpha + 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4e+15)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.4e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(2.0 * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. associate-+r+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. distribute-rgt1-in 99.9%

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

      8. +-commutative 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 80.7%

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

      1. +-commutative 80.7%

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

   7. Simplified 80.7%

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

   if 4.4e15 < beta

   1. Initial program 80.2%

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

      1. associate-/l/ 74.6%

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

      2. +-commutative 74.6%

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

      3. +-commutative 74.6%

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

      4. associate-+r+ 74.6%

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

      5. associate-+r+ 74.6%

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

      6. associate-+r+ 74.6%

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

      7. distribute-rgt1-in 74.6%

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

      8. +-commutative 74.6%

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

      9. *-commutative 74.6%

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

      10. distribute-rgt1-in 74.6%

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

      11. +-commutative 74.6%

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

      12. metadata-eval 74.6%

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

      13. associate-+l+ 74.6%

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

   3. Simplified 74.6%

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

      1. associate-+r+ 74.6%

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

      2. metadata-eval 74.6%

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

      3. associate-/l* 85.8%

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

      4. associate-+r+ 85.8%

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

      5. metadata-eval 85.8%

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

      6. associate-+r+ 85.8%

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

      7. metadata-eval 85.8%

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

      8. associate-+l+ 85.8%

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

      9. metadata-eval 85.8%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. metadata-eval 88.8%

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

      3. distribute-lft-in 88.8%

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

   9. Simplified 88.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-/l/ 91.9%

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

   2. +-commutative 91.9%

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

   3. +-commutative 91.9%

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

   4. associate-+r+ 91.9%

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

   5. associate-+r+ 91.9%

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

   6. associate-+r+ 91.9%

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

   7. distribute-rgt1-in 91.9%

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

   8. +-commutative 91.9%

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

   9. *-commutative 91.9%

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

   10. distribute-rgt1-in 91.9%

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

   11. +-commutative 91.9%

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

   12. metadata-eval 91.9%

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

   13. associate-+l+ 91.9%

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

3. Simplified 91.9%

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

   1. associate-+r+ 91.9%

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

   2. metadata-eval 91.9%

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

   3. associate-/l* 95.4%

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

   4. associate-+r+ 95.4%

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

   5. metadata-eval 95.4%

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

   6. associate-+r+ 95.4%

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

   7. metadata-eval 95.4%

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

   8. associate-+l+ 95.4%

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

   9. metadata-eval 95.4%

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

   10. times-frac 99.8%

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

   11. metadata-eval 99.8%

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

   12. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 5: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4e+16)
   (/
    (/ (+ 1.0 beta) (+ 2.0 beta))
    (* (+ alpha (+ 2.0 beta)) (+ alpha (+ beta 3.0))))
   (*
    (/ (+ alpha 1.0) (+ 2.0 (+ alpha beta)))
    (/ (- 1.0 (* 2.0 (/ alpha beta))) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (2.0 * (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 <= 4d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((alpha + (2.0d0 + beta)) * (alpha + (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (2.0d0 + (alpha + beta))) * ((1.0d0 - (2.0d0 * (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 <= 4e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4e+16:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)))
	else:
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(alpha + beta))) * Float64(Float64(1.0 - Float64(2.0 * Float64(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 <= 4e+16)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (2.0 * (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, 4e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e16

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. associate-+r+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. distribute-rgt1-in 99.9%

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

      8. +-commutative 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 80.8%

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

      1. +-commutative 80.8%

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

   7. Simplified 80.8%

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

   if 4e16 < beta

   1. Initial program 79.9%

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

      1. associate-/l/ 74.3%

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

      2. +-commutative 74.3%

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

      3. +-commutative 74.3%

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

      4. associate-+r+ 74.3%

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

      5. associate-+r+ 74.3%

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

      6. associate-+r+ 74.3%

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

      7. distribute-rgt1-in 74.3%

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

      8. +-commutative 74.3%

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

      9. *-commutative 74.3%

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

      10. distribute-rgt1-in 74.4%

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

      11. +-commutative 74.4%

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

      12. metadata-eval 74.4%

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

      13. associate-+l+ 74.4%

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

   3. Simplified 74.4%

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

      1. associate-+r+ 74.4%

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

      2. metadata-eval 74.4%

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

      3. associate-/l* 85.6%

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

      4. associate-+r+ 85.6%

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

      5. metadata-eval 85.6%

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

      6. associate-+r+ 85.6%

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

      7. metadata-eval 85.6%

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

      8. associate-+l+ 85.6%

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

      9. metadata-eval 85.6%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. metadata-eval 88.7%

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

      3. distribute-lft-in 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in alpha around inf 88.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))) (t_1 (+ alpha (+ beta 3.0))))
   (if (<= beta 1.2e+16)
     (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* t_0 t_1))
     (/ (/ (+ alpha 1.0) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 1.2e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_0 * t_1);
	} else {
		tmp = ((alpha + 1.0) / t_0) / t_1;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    t_1 = alpha + (beta + 3.0d0)
    if (beta <= 1.2d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (t_0 * t_1)
    else
        tmp = ((alpha + 1.0d0) / t_0) / t_1
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 1.2e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_0 * t_1);
	} else {
		tmp = ((alpha + 1.0) / t_0) / t_1;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	t_1 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 1.2e+16:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_0 * t_1)
	else:
		tmp = ((alpha + 1.0) / t_0) / t_1
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	t_1 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 1.2e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / t_1);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	t_1 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 1.2e+16)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_0 * t_1);
	else
		tmp = ((alpha + 1.0) / t_0) / t_1;
	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[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.2e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.2e16

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. associate-+r+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. distribute-rgt1-in 99.9%

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

      8. +-commutative 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 80.8%

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

      1. +-commutative 80.8%

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

   7. Simplified 80.8%

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

   if 1.2e16 < beta

   1. Initial program 79.9%

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

      1. associate-/l/ 74.3%

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

      2. +-commutative 74.3%

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

      3. +-commutative 74.3%

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

      4. associate-+r+ 74.3%

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

      5. associate-+r+ 74.3%

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

      6. associate-+r+ 74.3%

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

      7. distribute-rgt1-in 74.3%

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

      8. +-commutative 74.3%

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

      9. *-commutative 74.3%

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

      10. distribute-rgt1-in 74.4%

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

      11. +-commutative 74.4%

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

      12. metadata-eval 74.4%

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

      13. associate-+l+ 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in beta around inf 84.4%

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

      1. associate-+r+ 84.4%

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

      2. associate-/r* 89.3%

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

      3. +-commutative 89.3%

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

      4. +-commutative 89.3%

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

      5. associate-+r+ 89.3%

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

   7. Applied egg-rr 89.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/
    (/ (+ alpha 1.0) (+ alpha 2.0))
    (* (+ alpha (+ 2.0 beta)) (+ alpha (+ beta 3.0))))
   (* (/ (+ alpha 1.0) (+ 2.0 (+ alpha beta))) (/ (- 1.0 (/ 4.0 beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (4.0 / 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 <= 9.0d0) then
        tmp = ((alpha + 1.0d0) / (alpha + 2.0d0)) / ((alpha + (2.0d0 + beta)) * (alpha + (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (2.0d0 + (alpha + beta))) * ((1.0d0 - (4.0d0 / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.0:
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)))
	else:
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (4.0 / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + 2.0)) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(alpha + beta))) * Float64(Float64(1.0 - Float64(4.0 / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.0)
		tmp = ((alpha + 1.0) / (alpha + 2.0)) / ((alpha + (2.0 + beta)) * (alpha + (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) * ((1.0 - (4.0 / 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, 9.0], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(4.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. associate-+r+ 99.9%

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

      6. associate-+r+ 99.9%

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

      7. distribute-rgt1-in 99.9%

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

      8. +-commutative 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

   if 9 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

      1. associate-+r+ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. metadata-eval 86.7%

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

      6. associate-+r+ 86.7%

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

      7. metadata-eval 86.7%

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

      8. associate-+l+ 86.7%

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

      9. metadata-eval 86.7%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. metadata-eval 86.5%

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

      3. distribute-lft-in 86.5%

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

   9. Simplified 86.5%

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

   10. Taylor expanded in alpha around 0 86.5%

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

       1. associate-*r/ 86.5%

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

       2. metadata-eval 86.5%

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

   12. Simplified 86.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{0.25}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 5.8)
     (/ 0.25 (+ 1.0 t_0))
     (* (/ (+ alpha 1.0) t_0) (/ (- 1.0 (/ 4.0 beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 5.8) {
		tmp = 0.25 / (1.0 + t_0);
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 5.8d0) then
        tmp = 0.25d0 / (1.0d0 + t_0)
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - (4.0d0 / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 5.8) {
		tmp = 0.25 / (1.0 + t_0);
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 5.8:
		tmp = 0.25 / (1.0 + t_0)
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (4.0 / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 5.8)
		tmp = Float64(0.25 / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(4.0 / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 5.8)
		tmp = 0.25 / (1.0 + t_0);
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (4.0 / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.8], N[(0.25 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(4.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999982

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 5.79999999999999982 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

      1. associate-+r+ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. metadata-eval 86.7%

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

      6. associate-+r+ 86.7%

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

      7. metadata-eval 86.7%

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

      8. associate-+l+ 86.7%

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

      9. metadata-eval 86.7%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. metadata-eval 86.5%

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

      3. distribute-lft-in 86.5%

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

   9. Simplified 86.5%

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

   10. Taylor expanded in alpha around 0 86.5%

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

       1. associate-*r/ 86.5%

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

       2. metadata-eval 86.5%

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

   12. Simplified 86.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ alpha 1.0) (+ alpha (+ 2.0 beta))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (2.0d0 + beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(2.0 + beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 2 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in beta around inf 83.2%

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

      1. associate-+r+ 83.2%

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

      2. associate-/r* 86.6%

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

      3. +-commutative 86.6%

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

      4. +-commutative 86.6%

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

      5. associate-+r+ 86.6%

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

   7. Applied egg-rr 86.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ alpha 1.0) (+ alpha (+ 2.0 beta))) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (2.0d0 + beta))) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(2.0 + beta))) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 2.10000000000000009 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in beta around inf 83.2%

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

      1. associate-+r+ 83.2%

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

      2. associate-/r* 86.6%

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

      3. +-commutative 86.6%

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

      4. +-commutative 86.6%

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

      5. associate-+r+ 86.6%

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

   7. Applied egg-rr 86.6%

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

   8. Taylor expanded in alpha around 0 86.0%

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

      1. +-commutative 86.0%

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

   10. Simplified 86.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.3)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.3d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.3:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.3)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.3)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.3], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.29999999999999982

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.29999999999999982 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in beta around inf 83.2%

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

      1. associate-+r+ 83.2%

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

      2. associate-/r* 86.6%

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

      3. +-commutative 86.6%

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

      4. +-commutative 86.6%

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

      5. associate-+r+ 86.6%

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

   7. Applied egg-rr 86.6%

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

   8. Taylor expanded in beta around inf 85.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ alpha 1.0) (+ alpha (+ 2.0 beta))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.8d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (2.0d0 + beta))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(2.0 + beta))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / (alpha + (2.0 + beta))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.79999999999999982 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

      1. associate-+r+ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. metadata-eval 86.7%

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

      6. associate-+r+ 86.7%

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

      7. metadata-eval 86.7%

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

      8. associate-+l+ 86.7%

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

      9. metadata-eval 86.7%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. un-div-inv 85.8%

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

      3. +-commutative 85.8%

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

      4. +-commutative 85.8%

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

   9. Applied egg-rr 85.8%

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

       1. *-lft-identity 85.8%

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

       2. associate-+r+ 85.8%

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

       3. +-commutative 85.8%

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

   11. Simplified 85.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (* (/ (+ alpha 1.0) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.6d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.6], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 6.5999999999999996 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

      1. associate-+r+ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. metadata-eval 86.7%

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

      6. associate-+r+ 86.7%

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

      7. metadata-eval 86.7%

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

      8. associate-+l+ 86.7%

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

      9. metadata-eval 86.7%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 85.7%

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

   8. Taylor expanded in beta around inf 85.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (* (/ (+ alpha 1.0) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((alpha + 1.0) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   7. Taylor expanded in beta around 0 63.3%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   8. Step-by-step derivation

      1. *-commutative 63.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

      1. associate-+r+ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. metadata-eval 86.7%

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

      6. associate-+r+ 86.7%

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

      7. metadata-eval 86.7%

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

      8. associate-+l+ 86.7%

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

      9. metadata-eval 86.7%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 85.7%

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

   8. Taylor expanded in beta around inf 85.5%

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

Alternative 15: 92.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   7. Taylor expanded in beta around 0 63.3%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   8. Step-by-step derivation

      1. *-commutative 63.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.60000000000000009 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-+r+ 76.4%

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

      5. associate-+r+ 76.4%

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

      6. associate-+r+ 76.4%

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

      7. distribute-rgt1-in 76.4%

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

      8. +-commutative 76.4%

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

      9. *-commutative 76.4%

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

      10. distribute-rgt1-in 76.4%

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

      11. +-commutative 76.4%

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

      12. metadata-eval 76.4%

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

      13. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

      1. associate-+r+ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-/l* 86.7%

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

      4. associate-+r+ 86.7%

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

      5. metadata-eval 86.7%

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

      6. associate-+r+ 86.7%

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

      7. metadata-eval 86.7%

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

      8. associate-+l+ 86.7%

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

      9. metadata-eval 86.7%

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

      10. times-frac 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around inf 85.7%

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

   8. Taylor expanded in alpha around 0 73.5%

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

Alternative 16: 47.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / 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 <= 2.7d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.25 / 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, 2.7], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   7. Taylor expanded in beta around 0 63.3%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   8. Step-by-step derivation

      1. *-commutative 63.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7000000000000002 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around 0 17.2%

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

   4. Taylor expanded in alpha around 0 7.1%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   7. Taylor expanded in beta around inf 7.1%

      \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0) 0.08333333333333333 (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.25 / 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 <= 3.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.25 / 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, 3.0], 0.08333333333333333, N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.9%

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

   4. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   7. Taylor expanded in beta around 0 63.2%

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

   if 3 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around 0 17.2%

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

   4. Taylor expanded in alpha around 0 7.1%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   7. Taylor expanded in beta around inf 7.1%

      \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (beta + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around 0 71.8%

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

4. Taylor expanded in alpha around 0 44.2%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
5. Step-by-step derivation

   1. +-commutative 44.2%

      \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

6. Simplified 44.2%

   \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]
7. Add Preprocessing

Alternative 19: 45.1% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around 0 71.8%

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

4. Taylor expanded in alpha around 0 44.2%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
5. Step-by-step derivation

   1. +-commutative 44.2%

      \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

6. Simplified 44.2%

   \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

7. Taylor expanded in beta around 0 43.2%

   \[\leadsto \color{blue}{0.08333333333333333} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))