Octave 3.8, jcobi/3

Percentage Accurate: 93.8% → 99.7%

Time: 16.2s

Alternatives: 21

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: 93.8% 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

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{2 + \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\beta \cdot \left(\frac{-2 - \alpha}{\beta} + -1\right)} \cdot \frac{1 - \frac{2 \cdot \left(2 + \alpha\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4e+118)
   (/
    (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (+ 2.0 (+ alpha beta)))
    (* (+ (+ alpha beta) 3.0) (+ alpha (+ 2.0 beta))))
   (*
    (/ (- -1.0 alpha) (* beta (+ (/ (- -2.0 alpha) beta) -1.0)))
    (/ (- 1.0 (/ (* 2.0 (+ 2.0 alpha)) beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4e+118) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * (alpha + (2.0 + beta)));
	} else {
		tmp = ((-1.0 - alpha) / (beta * (((-2.0 - alpha) / beta) + -1.0))) * ((1.0 - ((2.0 * (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 <= 3.4d+118) then
        tmp = (((1.0d0 + beta) * (1.0d0 + alpha)) / (2.0d0 + (alpha + beta))) / (((alpha + beta) + 3.0d0) * (alpha + (2.0d0 + beta)))
    else
        tmp = (((-1.0d0) - alpha) / (beta * ((((-2.0d0) - alpha) / beta) + (-1.0d0)))) * ((1.0d0 - ((2.0d0 * (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 <= 3.4e+118) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * (alpha + (2.0 + beta)));
	} else {
		tmp = ((-1.0 - alpha) / (beta * (((-2.0 - alpha) / beta) + -1.0))) * ((1.0 - ((2.0 * (2.0 + alpha)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.4e+118:
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * (alpha + (2.0 + beta)))
	else:
		tmp = ((-1.0 - alpha) / (beta * (((-2.0 - alpha) / beta) + -1.0))) * ((1.0 - ((2.0 * (2.0 + alpha)) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4e+118)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(2.0 + Float64(alpha + beta))) / Float64(Float64(Float64(alpha + beta) + 3.0) * Float64(alpha + Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) / Float64(beta * Float64(Float64(Float64(-2.0 - alpha) / beta) + -1.0))) * Float64(Float64(1.0 - Float64(Float64(2.0 * Float64(2.0 + alpha)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.4e+118)
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * (alpha + (2.0 + beta)));
	else
		tmp = ((-1.0 - alpha) / (beta * (((-2.0 - alpha) / beta) + -1.0))) * ((1.0 - ((2.0 * (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, 3.4e+118], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / N[(beta * N[(N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(2.0 * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999986e118

   1. Initial program 99.4%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. +-commutative 99.2%

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

      2. associate-+r+ 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. metadata-eval 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. +-commutative 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-+r+ 99.2%

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

      10. +-commutative 99.2%

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

      11. distribute-rgt1-in 99.3%

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

      12. fma-define 99.3%

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

      13. metadata-eval 99.3%

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

      14. associate-+r+ 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. *-lft-identity 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-undefine 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. associate-+r+ 99.2%

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

      8. distribute-lft1-in 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. associate-+r+ 99.3%

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

      12. +-commutative 99.3%

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

      13. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   if 3.39999999999999986e118 < beta

   1. Initial program 64.0%

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

   3. Simplified 51.4%

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

      1. times-frac 86.0%

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

      2. +-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in beta around inf 94.4%

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

      1. mul-1-neg 94.4%

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

      2. metadata-eval 94.4%

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

      3. distribute-lft-in 94.4%

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

   9. Simplified 94.4%

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

   10. Taylor expanded in beta around -inf 94.4%

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

       1. associate-*r* 94.4%

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

       2. mul-1-neg 94.4%

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

       3. sub-neg 94.4%

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

       4. associate-*r/ 94.4%

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

       5. distribute-lft-in 94.4%

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

       6. metadata-eval 94.4%

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

       7. mul-1-neg 94.4%

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

       8. unsub-neg 94.4%

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

       9. metadata-eval 94.4%

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

   12. Simplified 94.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{2 + \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \alpha}{\beta \cdot \left(\frac{-2 - \alpha}{\beta} + -1\right)} \cdot \frac{1 - \frac{2 \cdot \left(2 + \alpha\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{2 + \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0} \cdot \frac{1 - \frac{2 \cdot \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 2e+118)
     (/
      (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (+ 2.0 (+ alpha beta)))
      (* (+ (+ alpha beta) 3.0) t_0))
     (* (/ (+ 1.0 alpha) t_0) (/ (- 1.0 (/ (* 2.0 alpha) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2e+118) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / 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 = alpha + (2.0d0 + beta)
    if (beta <= 2d+118) then
        tmp = (((1.0d0 + beta) * (1.0d0 + alpha)) / (2.0d0 + (alpha + beta))) / (((alpha + beta) + 3.0d0) * t_0)
    else
        tmp = ((1.0d0 + alpha) / 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 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2e+118) {
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((2.0 * alpha) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 2e+118:
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * t_0)
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 - ((2.0 * alpha) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 2e+118)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(2.0 + Float64(alpha + beta))) / Float64(Float64(Float64(alpha + beta) + 3.0) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 - Float64(Float64(2.0 * alpha) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 2e+118)
		tmp = (((1.0 + beta) * (1.0 + alpha)) / (2.0 + (alpha + beta))) / (((alpha + beta) + 3.0) * t_0);
	else
		tmp = ((1.0 + alpha) / 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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+118], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(2.0 * alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999993e118

   1. Initial program 99.4%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

      1. +-commutative 99.2%

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

      2. associate-+r+ 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-+r+ 99.2%

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

      5. metadata-eval 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. +-commutative 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-+r+ 99.2%

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

      10. +-commutative 99.2%

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

      11. distribute-rgt1-in 99.3%

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

      12. fma-define 99.3%

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

      13. metadata-eval 99.3%

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

      14. associate-+r+ 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. *-lft-identity 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-undefine 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. +-commutative 99.2%

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

      7. associate-+r+ 99.2%

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

      8. distribute-lft1-in 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. associate-+r+ 99.3%

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

      12. +-commutative 99.3%

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

      13. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   if 1.99999999999999993e118 < beta

   1. Initial program 64.0%

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

   3. Simplified 51.4%

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

      1. times-frac 86.0%

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

      2. +-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in beta around inf 94.4%

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

      1. mul-1-neg 94.4%

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

      2. metadata-eval 94.4%

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

      3. distribute-lft-in 94.4%

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

   9. Simplified 94.4%

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

   10. Taylor expanded in alpha around inf 94.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{2 + \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := \frac{1 + \alpha}{t\_0}\\ \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+96}:\\ \;\;\;\;t\_1 \cdot \frac{1 + \beta}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1 - \frac{2 \cdot \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))) (t_1 (/ (+ 1.0 alpha) t_0)))
   (if (<= beta 1.15e+96)
     (* t_1 (/ (+ 1.0 beta) (* t_0 (+ alpha (+ beta 3.0)))))
     (* t_1 (/ (- 1.0 (/ (* 2.0 alpha) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double t_1 = (1.0 + alpha) / t_0;
	double tmp;
	if (beta <= 1.15e+96) {
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = t_1 * ((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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    t_1 = (1.0d0 + alpha) / t_0
    if (beta <= 1.15d+96) then
        tmp = t_1 * ((1.0d0 + beta) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = t_1 * ((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 = alpha + (2.0 + beta);
	double t_1 = (1.0 + alpha) / t_0;
	double tmp;
	if (beta <= 1.15e+96) {
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = t_1 * ((1.0 - ((2.0 * alpha) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	t_1 = (1.0 + alpha) / t_0
	tmp = 0
	if beta <= 1.15e+96:
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = t_1 * ((1.0 - ((2.0 * alpha) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	t_1 = Float64(Float64(1.0 + alpha) / t_0)
	tmp = 0.0
	if (beta <= 1.15e+96)
		tmp = Float64(t_1 * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 - Float64(Float64(2.0 * alpha) / beta)) / beta));
	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 = (1.0 + alpha) / t_0;
	tmp = 0.0;
	if (beta <= 1.15e+96)
		tmp = t_1 * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = t_1 * ((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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 1.15e+96], N[(t$95$1 * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 - N[(N[(2.0 * alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15000000000000008e96

   1. Initial program 99.4%

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

   3. Simplified 94.1%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 1.15000000000000008e96 < beta

   1. Initial program 67.7%

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

   3. Simplified 48.3%

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

      1. times-frac 87.3%

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

      2. +-commutative 87.3%

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

   6. Applied egg-rr 87.3%

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

   7. Taylor expanded in beta around inf 94.8%

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

      1. mul-1-neg 94.8%

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

      2. metadata-eval 94.8%

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

      3. distribute-lft-in 94.8%

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

   9. Simplified 94.8%

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

   10. Taylor expanded in alpha around inf 94.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+96}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. associate-/l/ 91.7%

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

   2. +-commutative 91.7%

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

   3. associate-+l+ 91.7%

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

   4. *-commutative 91.7%

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

   5. metadata-eval 91.7%

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

   6. associate-+l+ 91.7%

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

   7. metadata-eval 91.7%

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

   8. +-commutative 91.7%

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

   9. +-commutative 91.7%

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

   10. +-commutative 91.7%

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

   11. metadata-eval 91.7%

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

   12. metadata-eval 91.7%

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

   13. associate-+l+ 91.7%

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

3. Simplified 91.7%

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

   1. +-commutative 91.7%

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

   2. associate-+r+ 91.7%

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

   3. *-commutative 91.7%

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

   4. associate-+r+ 91.7%

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

   5. metadata-eval 91.7%

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

   6. *-un-lft-identity 91.7%

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

   7. +-commutative 91.7%

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

   8. *-commutative 91.7%

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

   9. associate-+r+ 91.7%

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

   10. +-commutative 91.7%

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

   11. distribute-rgt1-in 91.7%

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

   12. fma-define 91.7%

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

   13. metadata-eval 91.7%

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

   14. associate-+r+ 91.7%

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

6. Applied egg-rr 91.7%

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

   1. *-lft-identity 91.7%

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

   2. +-commutative 91.7%

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

   3. fma-undefine 91.7%

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

   4. +-commutative 91.7%

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

   5. *-commutative 91.7%

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

   6. +-commutative 91.7%

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

   7. associate-+r+ 91.7%

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

   8. distribute-lft1-in 91.7%

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

   9. +-commutative 91.7%

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

   10. +-commutative 91.7%

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

   11. associate-+r+ 91.7%

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

   12. +-commutative 91.7%

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

   13. +-commutative 91.7%

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

8. Simplified 91.7%

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

   1. clear-num 91.7%

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

   2. inv-pow 91.7%

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

   3. *-commutative 91.7%

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

   4. associate-+r+ 91.7%

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

   5. associate-/l* 96.9%

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

   6. associate-+r+ 96.9%

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

   7. +-commutative 96.9%

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

   8. +-commutative 96.9%

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

10. Applied egg-rr 96.9%

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

    1. unpow-1 96.9%

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

    2. associate-/l* 99.4%

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

    3. associate-+r+ 99.4%

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

    4. +-commutative 99.4%

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

    5. +-commutative 99.4%

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

    6. +-commutative 99.4%

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

    7. associate-+r+ 99.4%

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

    8. associate-+r+ 99.4%

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

    9. +-commutative 99.4%

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

    10. +-commutative 99.4%

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

    11. +-commutative 99.4%

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

12. Simplified 99.4%

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

Alternative 5: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{2 \cdot \alpha + 4}{\beta}}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8e+62)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ (+ alpha beta) 3.0) (+ 2.0 beta)))
   (/
    (* (+ 1.0 alpha) (/ (- 1.0 (/ (+ (* 2.0 alpha) 4.0) beta)) beta))
    (+ alpha (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta)) / (alpha + (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 <= 3.8d+62) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (((alpha + beta) + 3.0d0) * (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 - (((2.0d0 * alpha) + 4.0d0) / beta)) / beta)) / (alpha + (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 <= 3.8e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta)) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8e+62:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta)) / (alpha + (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8e+62)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(Float64(alpha + beta) + 3.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 - Float64(Float64(Float64(2.0 * alpha) + 4.0) / beta)) / beta)) / Float64(alpha + 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 <= 3.8e+62)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	else
		tmp = ((1.0 + alpha) * ((1.0 - (((2.0 * alpha) + 4.0) / beta)) / beta)) / (alpha + (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, 3.8e+62], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(2.0 * alpha), $MachinePrecision] + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.79999999999999984e62

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in alpha around 0 71.2%

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

      1. +-commutative 71.2%

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

   10. Simplified 71.2%

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

   if 3.79999999999999984e62 < beta

   1. Initial program 70.1%

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

   3. Simplified 50.1%

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

      1. times-frac 88.8%

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

      2. +-commutative 88.8%

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

   6. Applied egg-rr 88.8%

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

   7. Taylor expanded in beta around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. metadata-eval 90.6%

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

      3. distribute-lft-in 90.6%

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

   9. Simplified 90.6%

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

       1. associate-*l/ 90.6%

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

       2. unsub-neg 90.6%

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

       3. distribute-lft-in 90.6%

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

       4. metadata-eval 90.6%

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

   11. Applied egg-rr 90.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{2 \cdot \alpha + 4}{\beta}}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+70)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ (+ alpha beta) 3.0) (+ 2.0 beta)))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (/ (- 1.0 (/ (* 2.0 alpha) beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+70) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + 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 <= 2d+70) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (((alpha + beta) + 3.0d0) * (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + 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 <= 2e+70) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * ((1.0 - ((2.0 * alpha) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+70:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + 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 <= 2e+70)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(Float64(alpha + beta) + 3.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(Float64(1.0 - Float64(Float64(2.0 * alpha) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+70)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + 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, 2e+70], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(2.0 * alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000015e70

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 87.7%

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

      1. +-commutative 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in alpha around 0 70.9%

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

      1. +-commutative 70.9%

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

   10. Simplified 70.9%

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

   if 2.00000000000000015e70 < beta

   1. Initial program 69.6%

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

   3. Simplified 50.8%

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

      1. times-frac 88.6%

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

      2. +-commutative 88.6%

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

   6. Applied egg-rr 88.6%

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

   7. Taylor expanded in beta around inf 91.9%

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

      1. mul-1-neg 91.9%

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

      2. metadata-eval 91.9%

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

      3. distribute-lft-in 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in alpha around inf 91.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 - \frac{2 \cdot \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta + \left(2 + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5e+62)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ (+ alpha beta) 3.0) (+ 2.0 beta)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (+ beta (+ 2.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d+62) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (((alpha + beta) + 3.0d0) * (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / (beta + (2.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5e+62:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5e+62)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(Float64(alpha + beta) + 3.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / Float64(beta + Float64(2.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5e+62)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((alpha + beta) + 3.0) * (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5e+62], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.49999999999999999e62

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in alpha around 0 71.2%

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

      1. +-commutative 71.2%

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

   10. Simplified 71.2%

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

   if 4.49999999999999999e62 < beta

   1. Initial program 70.1%

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

      1. associate-/l/ 68.5%

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

      2. +-commutative 68.5%

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

      3. associate-+l+ 68.5%

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

      4. *-commutative 68.5%

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

      5. metadata-eval 68.5%

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

      6. associate-+l+ 68.5%

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

      7. metadata-eval 68.5%

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

      8. +-commutative 68.5%

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

      9. +-commutative 68.5%

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

      10. +-commutative 68.5%

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

      11. metadata-eval 68.5%

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

      12. metadata-eval 68.5%

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

      13. associate-+l+ 68.5%

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

   3. Simplified 68.5%

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

      1. +-commutative 68.5%

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

      2. associate-+r+ 68.5%

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

      3. *-commutative 68.5%

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

      4. associate-+r+ 68.5%

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

      5. metadata-eval 68.5%

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

      6. *-un-lft-identity 68.5%

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

      7. +-commutative 68.5%

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

      8. *-commutative 68.5%

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

      9. associate-+r+ 68.5%

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

      10. +-commutative 68.5%

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

      11. distribute-rgt1-in 68.5%

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

      12. fma-define 68.5%

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

      13. metadata-eval 68.5%

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

      14. associate-+r+ 68.5%

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

   6. Applied egg-rr 68.5%

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

      1. *-lft-identity 68.5%

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

      2. +-commutative 68.5%

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

      3. fma-undefine 68.5%

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

      4. +-commutative 68.5%

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

      5. *-commutative 68.5%

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

      6. +-commutative 68.5%

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

      7. associate-+r+ 68.5%

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

      8. distribute-lft1-in 68.5%

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

      9. +-commutative 68.5%

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

      10. +-commutative 68.5%

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

      11. associate-+r+ 68.5%

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

      12. +-commutative 68.5%

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

      13. +-commutative 68.5%

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

   8. Simplified 68.5%

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

   9. Taylor expanded in beta around inf 85.3%

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

       1. *-un-lft-identity 85.3%

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

       2. associate-/r* 90.7%

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

       3. associate-+l+ 90.7%

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

       4. associate-+r+ 90.7%

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

       5. +-commutative 90.7%

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

       6. associate-+r+ 90.7%

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

   11. Applied egg-rr 90.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta + \left(2 + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta + \left(2 + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+62)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (+ beta (+ 2.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+62) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / (beta + (2.0d0 + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5e+62:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+62)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / Float64(beta + Float64(2.0 + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+62)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (beta + (2.0 + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5e+62], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000029e62

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in alpha around 0 69.9%

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

   if 5.00000000000000029e62 < beta

   1. Initial program 70.1%

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

      1. associate-/l/ 68.5%

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

      2. +-commutative 68.5%

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

      3. associate-+l+ 68.5%

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

      4. *-commutative 68.5%

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

      5. metadata-eval 68.5%

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

      6. associate-+l+ 68.5%

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

      7. metadata-eval 68.5%

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

      8. +-commutative 68.5%

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

      9. +-commutative 68.5%

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

      10. +-commutative 68.5%

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

      11. metadata-eval 68.5%

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

      12. metadata-eval 68.5%

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

      13. associate-+l+ 68.5%

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

   3. Simplified 68.5%

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

      1. +-commutative 68.5%

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

      2. associate-+r+ 68.5%

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

      3. *-commutative 68.5%

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

      4. associate-+r+ 68.5%

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

      5. metadata-eval 68.5%

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

      6. *-un-lft-identity 68.5%

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

      7. +-commutative 68.5%

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

      8. *-commutative 68.5%

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

      9. associate-+r+ 68.5%

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

      10. +-commutative 68.5%

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

      11. distribute-rgt1-in 68.5%

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

      12. fma-define 68.5%

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

      13. metadata-eval 68.5%

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

      14. associate-+r+ 68.5%

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

   6. Applied egg-rr 68.5%

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

      1. *-lft-identity 68.5%

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

      2. +-commutative 68.5%

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

      3. fma-undefine 68.5%

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

      4. +-commutative 68.5%

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

      5. *-commutative 68.5%

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

      6. +-commutative 68.5%

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

      7. associate-+r+ 68.5%

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

      8. distribute-lft1-in 68.5%

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

      9. +-commutative 68.5%

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

      10. +-commutative 68.5%

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

      11. associate-+r+ 68.5%

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

      12. +-commutative 68.5%

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

      13. +-commutative 68.5%

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

   8. Simplified 68.5%

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

   9. Taylor expanded in beta around inf 85.3%

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

       1. *-un-lft-identity 85.3%

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

       2. associate-/r* 90.7%

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

       3. associate-+l+ 90.7%

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

       4. associate-+r+ 90.7%

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

       5. +-commutative 90.7%

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

       6. associate-+r+ 90.7%

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

   11. Applied egg-rr 90.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\beta + \left(2 + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6e+62)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (* beta (- (/ (- alpha -3.0) beta) -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	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 <= 3.6d+62) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta * (((alpha - (-3.0d0)) / beta) - (-1.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6e+62:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6e+62)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta * Float64(Float64(Float64(alpha - -3.0) / beta) - -1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6e+62)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (beta * (((alpha - -3.0) / beta) - -1.0));
	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, 3.6e+62], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta * N[(N[(N[(alpha - -3.0), $MachinePrecision] / beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6e62

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in alpha around 0 69.9%

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

   if 3.6e62 < beta

   1. Initial program 70.1%

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

   3. Taylor expanded in beta around inf 90.5%

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

   4. Taylor expanded in beta around -inf 90.5%

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

      1. associate-*r* 90.5%

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

      2. mul-1-neg 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*r/ 90.5%

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

      5. distribute-lft-in 90.5%

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

      6. metadata-eval 90.5%

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

      7. metadata-eval 90.5%

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

      8. mul-1-neg 90.5%

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

      9. unsub-neg 90.5%

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

      10. metadata-eval 90.5%

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

      11. metadata-eval 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta \cdot \left(\frac{\alpha - -3}{\beta} - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4e+62)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 6.4d+62) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / 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 <= 6.4e+62) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4e+62:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4e+62)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + 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 <= 6.4e+62)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / 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, 6.4e+62], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.39999999999999968e62

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in alpha around 0 69.9%

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

   if 6.39999999999999968e62 < beta

   1. Initial program 70.1%

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

   3. Taylor expanded in beta around inf 90.5%

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

      1. *-un-lft-identity 90.5%

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

      2. metadata-eval 90.5%

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

      3. associate-+l+ 90.5%

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

      4. metadata-eval 90.5%

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

      5. associate-+r+ 90.5%

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

   5. Applied egg-rr 90.5%

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

      1. *-lft-identity 90.5%

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

      2. associate-+r+ 90.5%

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

   7. Simplified 90.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+
    0.08333333333333333
    (*
     alpha
     (-
      (* alpha (- (* alpha 0.024691358024691357) 0.011574074074074073))
      0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * ((alpha * 0.024691358024691357d0) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / 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 <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * Float64(Float64(alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + 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 = 0.08333333333333333 + (alpha * ((alpha * ((alpha * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / 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, 2.1], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * N[(N[(alpha * 0.024691358024691357), $MachinePrecision] - 0.011574074074074073), $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.9%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(0.024691358024691357 \cdot \alpha - 0.011574074074074073\right) - 0.027777777777777776\right)} \]

   if 2.10000000000000009 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

      1. *-un-lft-identity 84.5%

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

      2. metadata-eval 84.5%

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

      3. associate-+l+ 84.5%

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

      4. metadata-eval 84.5%

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

      5. associate-+r+ 84.5%

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

   5. Applied egg-rr 84.5%

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

      1. *-lft-identity 84.5%

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

      2. associate-+r+ 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot \left(\alpha \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.9)
   (+
    0.08333333333333333
    (* alpha (- (* alpha -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 1.9d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / 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 <= 1.9) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.9:
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.9)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + 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 <= 1.9)
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / 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, 1.9], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(-0.011574074074074073 \cdot \alpha - 0.027777777777777776\right)} \]

   if 1.8999999999999999 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

      1. *-un-lft-identity 84.5%

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

      2. metadata-eval 84.5%

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

      3. associate-+l+ 84.5%

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

      4. metadata-eval 84.5%

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

      5. associate-+r+ 84.5%

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

   5. Applied egg-rr 84.5%

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

      1. *-lft-identity 84.5%

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

      2. associate-+r+ 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (+
    0.08333333333333333
    (* alpha (- (* alpha -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * ((alpha * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2], N[(0.08333333333333333 + N[(alpha * N[(N[(alpha * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(-0.011574074074074073 \cdot \alpha - 0.027777777777777776\right)} \]

   if 2.2000000000000002 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

   4. Taylor expanded in alpha around 0 84.4%

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

      1. +-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+
    0.08333333333333333
    (* alpha (- (* alpha -0.011574074074074073) 0.027777777777777776)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / 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 = 0.08333333333333333d0 + (alpha * ((alpha * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = (1.0d0 / 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 = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = (1.0 / 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(0.08333333333333333 + Float64(alpha * Float64(Float64(alpha * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(1.0 / 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 = 0.08333333333333333 + (alpha * ((alpha * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = (1.0 / 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[(0.08333333333333333 + N[(alpha * N[(N[(alpha * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.4%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot \left(-0.011574074074074073 \cdot \alpha - 0.027777777777777776\right)} \]

   if 2.10000000000000009 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

   4. Taylor expanded in alpha around 0 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. associate-/r* 78.2%

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

      3. +-commutative 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. *-lft-identity 78.2%

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

   8. Simplified 78.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot \left(\alpha \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (1.0 / 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 = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / 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 = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = (1.0 / 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(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 / 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 = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = (1.0 / 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[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.3%

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

       1. *-commutative 68.3%

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

   13. Simplified 68.3%

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

   if 2.10000000000000009 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

   4. Taylor expanded in alpha around 0 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. associate-/r* 78.2%

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

      3. +-commutative 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. *-lft-identity 78.2%

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

   8. Simplified 78.2%

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

Alternative 16: 91.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.95)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (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 <= 1.95d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (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 <= 1.95) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.95:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.95)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(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 <= 1.95)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 1.0 / (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, 1.95], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.94999999999999996

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.3%

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

       1. *-commutative 68.3%

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

   13. Simplified 68.3%

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

   if 1.94999999999999996 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

   4. Taylor expanded in alpha around 0 77.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 91.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.9d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.9:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.9], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.3%

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

       1. *-commutative 68.3%

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

   13. Simplified 68.3%

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

   if 2.89999999999999991 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

   4. Taylor expanded in alpha around 0 77.4%

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

   5. Taylor expanded in beta around inf 77.3%

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

Alternative 18: 48.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 0.3333333333333333 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.5d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 0.3333333333333333d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.5:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 0.3333333333333333 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(0.3333333333333333 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.5)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 0.3333333333333333 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.5], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.3%

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

       1. *-commutative 68.3%

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

   13. Simplified 68.3%

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

   if 3.5 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

   4. Taylor expanded in alpha around 0 77.4%

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

   5. Taylor expanded in beta around 0 7.0%

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

Alternative 19: 47.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0) 0.08333333333333333 (/ 0.3333333333333333 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.3333333333333333d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.3333333333333333 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.3333333333333333 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.3333333333333333 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.3333333333333333 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.0], 0.08333333333333333, N[(0.3333333333333333 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.9%

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

   3. Simplified 95.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in beta around 0 98.0%

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

   11. Taylor expanded in alpha around 0 68.5%

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

   if 4 < beta

   1. Initial program 75.5%

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

   3. Taylor expanded in beta around inf 84.5%

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

   4. Taylor expanded in alpha around 0 77.4%

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

   5. Taylor expanded in beta around 0 7.0%

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

Alternative 20: 47.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

3. Simplified 83.7%

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

   1. times-frac 96.9%

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

   2. +-commutative 96.9%

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

6. Applied egg-rr 96.9%

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

7. Taylor expanded in beta around 0 75.2%

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

   1. +-commutative 75.2%

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

9. Simplified 75.2%

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

10. Taylor expanded in alpha around 0 49.1%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
11. Step-by-step derivation

    1. +-commutative 49.1%

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

12. Simplified 49.1%

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

13. Final simplification 49.1%

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

Alternative 21: 45.8% 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 92.2%

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

3. Simplified 83.7%

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

   1. times-frac 96.9%

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

   2. +-commutative 96.9%

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

6. Applied egg-rr 96.9%

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

7. Taylor expanded in beta around 0 75.2%

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

   1. +-commutative 75.2%

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

9. Simplified 75.2%

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

10. Taylor expanded in beta around 0 72.6%

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

11. Taylor expanded in alpha around 0 48.1%

    \[\leadsto \color{blue}{0.08333333333333333} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))