Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.5%

Time: 19.0s

Alternatives: 15

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \left({\left(\alpha + \left(2 + \beta\right)\right)}^{-2} \cdot \frac{1 + \beta}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= alpha 1.3e-26)
     (*
      (+ alpha 1.0)
      (* (pow (+ alpha (+ 2.0 beta)) -2.0) (/ (+ 1.0 beta) t_0)))
     (/
      (/
       (+
        (+ 1.0 (+ alpha (+ (/ 1.0 beta) (/ alpha beta))))
        (* (/ (+ 4.0 (* alpha 2.0)) beta) (- -1.0 alpha)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (alpha <= 1.3e-26) {
		tmp = (alpha + 1.0) * (pow((alpha + (2.0 + beta)), -2.0) * ((1.0 + beta) / t_0));
	} else {
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (alpha <= 1.3d-26) then
        tmp = (alpha + 1.0d0) * (((alpha + (2.0d0 + beta)) ** (-2.0d0)) * ((1.0d0 + beta) / t_0))
    else
        tmp = (((1.0d0 + (alpha + ((1.0d0 / beta) + (alpha / beta)))) + (((4.0d0 + (alpha * 2.0d0)) / beta) * ((-1.0d0) - alpha))) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (alpha <= 1.3e-26) {
		tmp = (alpha + 1.0) * (Math.pow((alpha + (2.0 + beta)), -2.0) * ((1.0 + beta) / t_0));
	} else {
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if alpha <= 1.3e-26:
		tmp = (alpha + 1.0) * (math.pow((alpha + (2.0 + beta)), -2.0) * ((1.0 + beta) / t_0))
	else:
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (alpha <= 1.3e-26)
		tmp = Float64(Float64(alpha + 1.0) * Float64((Float64(alpha + Float64(2.0 + beta)) ^ -2.0) * Float64(Float64(1.0 + beta) / t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(alpha + Float64(Float64(1.0 / beta) + Float64(alpha / beta)))) + Float64(Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta) * Float64(-1.0 - alpha))) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (alpha <= 1.3e-26)
		tmp = (alpha + 1.0) * (((alpha + (2.0 + beta)) ^ -2.0) * ((1.0 + beta) / t_0));
	else
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 1.3e-26], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[Power[N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(alpha + N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.30000000000000005e-26

   1. Initial program 99.8%

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

   3. Simplified 95.9%

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

      1. *-un-lft-identity 95.9%

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

      2. associate-/l* 95.9%

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

      3. +-commutative 95.9%

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

      4. associate-+r+ 95.9%

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

      5. associate-*r* 95.9%

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

      6. pow2 95.9%

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

      7. associate-+r+ 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. *-lft-identity 95.9%

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

      2. +-commutative 95.9%

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

      3. associate-/r* 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. associate-+r+ 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. div-inv 99.5%

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

      3. associate-+r+ 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. pow-flip 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-un-lft-identity 99.9%

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

       2. associate-/l* 99.9%

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

       3. +-commutative 99.9%

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

       4. *-commutative 99.9%

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

       5. +-commutative 99.9%

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

   12. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-/l* 99.9%

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

       3. +-commutative 99.9%

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

   14. Simplified 99.9%

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

   if 1.30000000000000005e-26 < alpha

   1. Initial program 83.8%

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

   3. Simplified 69.7%

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

      1. *-un-lft-identity 69.7%

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

      2. associate-/l* 80.6%

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

      3. +-commutative 80.6%

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

      4. associate-+r+ 80.6%

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

      5. associate-*r* 80.5%

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

      6. pow2 80.5%

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

      7. associate-+r+ 80.5%

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

   6. Applied egg-rr 80.5%

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

      1. *-lft-identity 80.5%

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

      2. +-commutative 80.5%

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

      3. associate-/r* 83.2%

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

      4. associate-+r+ 83.2%

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

      5. +-commutative 83.2%

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

      6. +-commutative 83.2%

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

      7. associate-+r+ 83.2%

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

      8. +-commutative 83.2%

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

      9. +-commutative 83.2%

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

   8. Simplified 83.2%

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

      1. associate-*r/ 92.6%

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

      2. div-inv 92.6%

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

      3. associate-+r+ 92.6%

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

      4. +-commutative 92.6%

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

      5. +-commutative 92.6%

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

      6. pow-flip 94.4%

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

      7. metadata-eval 94.4%

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

      8. associate-+r+ 94.4%

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

      9. +-commutative 94.4%

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

      10. +-commutative 94.4%

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

   10. Applied egg-rr 94.4%

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

   11. Taylor expanded in beta around inf 14.9%

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

       1. associate-/l* 17.5%

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

   13. Simplified 17.5%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \left({\left(\alpha + \left(2 + \beta\right)\right)}^{-2} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\beta \cdot \left(5 + \left(\beta + \alpha \cdot 2\right)\right) + \left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha \cdot \left(1 + \left(-2 \cdot \frac{\alpha}{\beta} - \frac{5}{\beta}\right)\right) - \frac{3}{\beta}\right)}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2e+56)
   (/
    (* (+ alpha 1.0) (+ 1.0 beta))
    (*
     (+ alpha (+ 2.0 beta))
     (+
      (* beta (+ 5.0 (+ beta (* alpha 2.0))))
      (* (+ alpha 2.0) (+ alpha 3.0)))))
   (/
    (/
     (+
      1.0
      (-
       (* alpha (+ 1.0 (- (* -2.0 (/ alpha beta)) (/ 5.0 beta))))
       (/ 3.0 beta)))
     beta)
    (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e+56) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))));
	} else {
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / 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 <= 1.2d+56) then
        tmp = ((alpha + 1.0d0) * (1.0d0 + beta)) / ((alpha + (2.0d0 + beta)) * ((beta * (5.0d0 + (beta + (alpha * 2.0d0)))) + ((alpha + 2.0d0) * (alpha + 3.0d0))))
    else
        tmp = ((1.0d0 + ((alpha * (1.0d0 + (((-2.0d0) * (alpha / beta)) - (5.0d0 / beta)))) - (3.0d0 / beta))) / 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 <= 1.2e+56) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))));
	} else {
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2e+56:
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))))
	else:
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2e+56)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 + beta)) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(beta * Float64(5.0 + Float64(beta + Float64(alpha * 2.0)))) + Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(alpha * Float64(1.0 + Float64(Float64(-2.0 * Float64(alpha / beta)) - Float64(5.0 / beta)))) - Float64(3.0 / beta))) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2e+56)
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))));
	else
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / 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, 1.2e+56], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * N[(5.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(alpha * N[(1.0 + N[(N[(-2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(5.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.20000000000000007e56

   1. Initial program 98.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in beta around 0 92.8%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   if 1.20000000000000007e56 < beta

   1. Initial program 75.7%

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

   3. Simplified 60.8%

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

      1. *-un-lft-identity 60.8%

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

      2. associate-/l* 76.3%

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

      3. +-commutative 76.3%

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

      4. associate-+r+ 76.3%

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

      5. associate-*r* 76.3%

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

      6. pow2 76.3%

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

      7. associate-+r+ 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. *-lft-identity 76.3%

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

      2. +-commutative 76.3%

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

      3. associate-/r* 88.4%

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

      4. associate-+r+ 88.4%

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

      5. +-commutative 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-+r+ 88.4%

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

      8. +-commutative 88.4%

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

      9. +-commutative 88.4%

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

   8. Simplified 88.4%

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

      1. associate-*r/ 88.4%

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

      2. div-inv 88.3%

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

      3. associate-+r+ 88.3%

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

      4. +-commutative 88.3%

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

      5. +-commutative 88.3%

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

      6. pow-flip 90.6%

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

      7. metadata-eval 90.6%

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

      8. associate-+r+ 90.6%

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

      9. +-commutative 90.6%

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

      10. +-commutative 90.6%

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

   10. Applied egg-rr 90.6%

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

   11. Taylor expanded in beta around inf 83.7%

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

       1. associate-/l* 88.2%

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

   13. Simplified 88.2%

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

   14. Taylor expanded in alpha around 0 88.2%

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

       1. associate--l+ 88.2%

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

       2. associate--l+ 88.2%

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

       3. associate-*r/ 88.2%

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

       4. metadata-eval 88.2%

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

       5. associate-*r/ 88.2%

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

       6. metadata-eval 88.2%

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

   16. Simplified 88.2%

       \[\leadsto \frac{\frac{\color{blue}{1 + \left(\alpha \cdot \left(1 + \left(-2 \cdot \frac{\alpha}{\beta} - \frac{5}{\beta}\right)\right) - \frac{3}{\beta}\right)}}{\beta}}{\alpha + \left(\beta + 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\beta \cdot \left(5 + \left(\beta + \alpha \cdot 2\right)\right) + \left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha \cdot \left(1 + \left(-2 \cdot \frac{\alpha}{\beta} - \frac{5}{\beta}\right)\right) - \frac{3}{\beta}\right)}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\beta \cdot \left(5 + \left(\beta + \alpha \cdot 2\right)\right) + \left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.25e+55)
   (/
    (* (+ alpha 1.0) (+ 1.0 beta))
    (*
     (+ alpha (+ 2.0 beta))
     (+
      (* beta (+ 5.0 (+ beta (* alpha 2.0))))
      (* (+ alpha 2.0) (+ alpha 3.0)))))
   (/
    (/
     (+
      (+ 1.0 (+ alpha (+ (/ 1.0 beta) (/ alpha beta))))
      (* (/ (+ 4.0 (* alpha 2.0)) beta) (- -1.0 alpha)))
     beta)
    (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.25e+55) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))));
	} else {
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-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 <= 1.25d+55) then
        tmp = ((alpha + 1.0d0) * (1.0d0 + beta)) / ((alpha + (2.0d0 + beta)) * ((beta * (5.0d0 + (beta + (alpha * 2.0d0)))) + ((alpha + 2.0d0) * (alpha + 3.0d0))))
    else
        tmp = (((1.0d0 + (alpha + ((1.0d0 / beta) + (alpha / beta)))) + (((4.0d0 + (alpha * 2.0d0)) / beta) * ((-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 <= 1.25e+55) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))));
	} else {
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.25e+55:
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))))
	else:
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-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 <= 1.25e+55)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 + beta)) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(beta * Float64(5.0 + Float64(beta + Float64(alpha * 2.0)))) + Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(alpha + Float64(Float64(1.0 / beta) + Float64(alpha / beta)))) + Float64(Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta) * Float64(-1.0 - alpha))) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.25e+55)
		tmp = ((alpha + 1.0) * (1.0 + beta)) / ((alpha + (2.0 + beta)) * ((beta * (5.0 + (beta + (alpha * 2.0)))) + ((alpha + 2.0) * (alpha + 3.0))));
	else
		tmp = (((1.0 + (alpha + ((1.0 / beta) + (alpha / beta)))) + (((4.0 + (alpha * 2.0)) / beta) * (-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, 1.25e+55], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * N[(5.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(alpha + N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25000000000000011e55

   1. Initial program 98.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in beta around 0 92.8%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   if 1.25000000000000011e55 < beta

   1. Initial program 75.7%

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

   3. Simplified 60.8%

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

      1. *-un-lft-identity 60.8%

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

      2. associate-/l* 76.3%

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

      3. +-commutative 76.3%

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

      4. associate-+r+ 76.3%

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

      5. associate-*r* 76.3%

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

      6. pow2 76.3%

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

      7. associate-+r+ 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. *-lft-identity 76.3%

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

      2. +-commutative 76.3%

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

      3. associate-/r* 88.4%

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

      4. associate-+r+ 88.4%

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

      5. +-commutative 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-+r+ 88.4%

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

      8. +-commutative 88.4%

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

      9. +-commutative 88.4%

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

   8. Simplified 88.4%

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

      1. associate-*r/ 88.4%

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

      2. div-inv 88.3%

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

      3. associate-+r+ 88.3%

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

      4. +-commutative 88.3%

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

      5. +-commutative 88.3%

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

      6. pow-flip 90.6%

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

      7. metadata-eval 90.6%

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

      8. associate-+r+ 90.6%

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

      9. +-commutative 90.6%

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

      10. +-commutative 90.6%

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

   10. Applied egg-rr 90.6%

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

   11. Taylor expanded in beta around inf 83.7%

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

       1. associate-/l* 88.2%

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

   13. Simplified 88.2%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\beta \cdot \left(5 + \left(\beta + \alpha \cdot 2\right)\right) + \left(\alpha + 2\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{t\_1 \cdot \left(t\_0 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha \cdot \left(1 + \left(-2 \cdot \frac{\alpha}{\beta} - \frac{5}{\beta}\right)\right) - \frac{3}{\beta}\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 5e+55)
     (/ (* (+ alpha 1.0) (+ 1.0 beta)) (* t_1 (* t_0 t_1)))
     (/
      (/
       (+
        1.0
        (-
         (* alpha (+ 1.0 (- (* -2.0 (/ alpha beta)) (/ 5.0 beta))))
         (/ 3.0 beta)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 5e+55) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 5d+55) then
        tmp = ((alpha + 1.0d0) * (1.0d0 + beta)) / (t_1 * (t_0 * t_1))
    else
        tmp = ((1.0d0 + ((alpha * (1.0d0 + (((-2.0d0) * (alpha / beta)) - (5.0d0 / beta)))) - (3.0d0 / beta))) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 5e+55) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 5e+55:
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1))
	else:
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 5e+55)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 + beta)) / Float64(t_1 * Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(alpha * Float64(1.0 + Float64(Float64(-2.0 * Float64(alpha / beta)) - Float64(5.0 / beta)))) - Float64(3.0 / beta))) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 5e+55)
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1));
	else
		tmp = ((1.0 + ((alpha * (1.0 + ((-2.0 * (alpha / beta)) - (5.0 / beta)))) - (3.0 / beta))) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+55], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(alpha * N[(1.0 + N[(N[(-2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(5.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000046e55

   1. Initial program 98.7%

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

   3. Simplified 92.7%

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

   if 5.00000000000000046e55 < beta

   1. Initial program 75.7%

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

   3. Simplified 60.8%

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

      1. *-un-lft-identity 60.8%

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

      2. associate-/l* 76.3%

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

      3. +-commutative 76.3%

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

      4. associate-+r+ 76.3%

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

      5. associate-*r* 76.3%

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

      6. pow2 76.3%

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

      7. associate-+r+ 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. *-lft-identity 76.3%

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

      2. +-commutative 76.3%

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

      3. associate-/r* 88.4%

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

      4. associate-+r+ 88.4%

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

      5. +-commutative 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-+r+ 88.4%

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

      8. +-commutative 88.4%

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

      9. +-commutative 88.4%

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

   8. Simplified 88.4%

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

      1. associate-*r/ 88.4%

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

      2. div-inv 88.3%

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

      3. associate-+r+ 88.3%

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

      4. +-commutative 88.3%

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

      5. +-commutative 88.3%

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

      6. pow-flip 90.6%

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

      7. metadata-eval 90.6%

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

      8. associate-+r+ 90.6%

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

      9. +-commutative 90.6%

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

      10. +-commutative 90.6%

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

   10. Applied egg-rr 90.6%

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

   11. Taylor expanded in beta around inf 83.7%

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

       1. associate-/l* 88.2%

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

   13. Simplified 88.2%

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

   14. Taylor expanded in alpha around 0 88.2%

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

       1. associate--l+ 88.2%

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

       2. associate--l+ 88.2%

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

       3. associate-*r/ 88.2%

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

       4. metadata-eval 88.2%

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

       5. associate-*r/ 88.2%

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

       6. metadata-eval 88.2%

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

   16. Simplified 88.2%

       \[\leadsto \frac{\frac{\color{blue}{1 + \left(\alpha \cdot \left(1 + \left(-2 \cdot \frac{\alpha}{\beta} - \frac{5}{\beta}\right)\right) - \frac{3}{\beta}\right)}}{\beta}}{\alpha + \left(\beta + 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha \cdot \left(1 + \left(-2 \cdot \frac{\alpha}{\beta} - \frac{5}{\beta}\right)\right) - \frac{3}{\beta}\right)}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 255000000:\\ \;\;\;\;\frac{\alpha + 1}{t\_1} \cdot \frac{1 + \beta}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 255000000.0)
     (* (/ (+ alpha 1.0) t_1) (/ (+ 1.0 beta) (* t_0 t_1)))
     (/
      (* (+ alpha 1.0) (/ (- 1.0 (/ (+ 3.0 (* alpha 2.0)) beta)) beta))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 255000000.0) {
		tmp = ((alpha + 1.0) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 255000000.0d0) then
        tmp = ((alpha + 1.0d0) / t_1) * ((1.0d0 + beta) / (t_0 * t_1))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - ((3.0d0 + (alpha * 2.0d0)) / beta)) / beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 255000000.0) {
		tmp = ((alpha + 1.0) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 255000000.0:
		tmp = ((alpha + 1.0) / t_1) * ((1.0 + beta) / (t_0 * t_1))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 255000000.0)
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(Float64(3.0 + Float64(alpha * 2.0)) / beta)) / beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 255000000.0)
		tmp = ((alpha + 1.0) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	else
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 255000000.0], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.55e8

   1. Initial program 99.8%

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

   3. Simplified 93.2%

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

      1. times-frac 99.3%

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

      2. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 2.55e8 < beta

   1. Initial program 78.6%

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

   3. Simplified 67.1%

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

      1. *-un-lft-identity 67.1%

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

      2. associate-/l* 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-*r* 81.5%

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

      6. pow2 81.5%

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

      7. associate-+r+ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-lft-identity 81.5%

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

      2. +-commutative 81.5%

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

      3. associate-/r* 90.9%

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

      4. associate-+r+ 90.9%

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

      5. +-commutative 90.9%

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

      6. +-commutative 90.9%

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

      7. associate-+r+ 90.9%

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

      8. +-commutative 90.9%

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

      9. +-commutative 90.9%

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

   8. Simplified 90.9%

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

      1. associate-*r/ 90.9%

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

      2. div-inv 90.8%

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

      3. associate-+r+ 90.8%

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

      4. +-commutative 90.8%

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

      5. +-commutative 90.8%

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

      6. pow-flip 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-+r+ 92.7%

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

      9. +-commutative 92.7%

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

      10. +-commutative 92.7%

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

   10. Applied egg-rr 92.7%

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

   11. Taylor expanded in beta around inf 84.5%

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

       1. mul-1-neg 84.5%

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

       2. +-commutative 84.5%

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

   13. Simplified 84.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 255000000:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{t\_1 \cdot \left(t\_0 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 1.65e+91)
     (/ (* (+ alpha 1.0) (+ 1.0 beta)) (* t_1 (* t_0 t_1)))
     (/
      (* (+ alpha 1.0) (/ (- 1.0 (/ (+ 3.0 (* alpha 2.0)) beta)) beta))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.65e+91) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 1.65d+91) then
        tmp = ((alpha + 1.0d0) * (1.0d0 + beta)) / (t_1 * (t_0 * t_1))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - ((3.0d0 + (alpha * 2.0d0)) / beta)) / beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.65e+91) {
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 1.65e+91:
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 1.65e+91)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 + beta)) / Float64(t_1 * Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(Float64(3.0 + Float64(alpha * 2.0)) / beta)) / beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 1.65e+91)
		tmp = ((alpha + 1.0) * (1.0 + beta)) / (t_1 * (t_0 * t_1));
	else
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.65e+91], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.65000000000000009e91

   1. Initial program 97.8%

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

   3. Simplified 92.0%

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

   if 1.65000000000000009e91 < beta

   1. Initial program 75.8%

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

   3. Simplified 58.7%

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

      1. *-un-lft-identity 58.7%

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

      2. associate-/l* 72.8%

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

      3. +-commutative 72.8%

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

      4. associate-+r+ 72.8%

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

      5. associate-*r* 72.8%

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

      6. pow2 72.8%

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

      7. associate-+r+ 72.8%

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

   6. Applied egg-rr 72.8%

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

      1. *-lft-identity 72.8%

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

      2. +-commutative 72.8%

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

      3. associate-/r* 86.8%

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

      4. associate-+r+ 86.8%

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

      5. +-commutative 86.8%

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

      6. +-commutative 86.8%

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

      7. associate-+r+ 86.8%

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

      8. +-commutative 86.8%

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

      9. +-commutative 86.8%

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

   8. Simplified 86.8%

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

      1. associate-*r/ 86.7%

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

      2. div-inv 86.7%

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

      3. associate-+r+ 86.7%

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

      4. +-commutative 86.7%

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

      5. +-commutative 86.7%

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

      6. pow-flip 89.3%

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

      7. metadata-eval 89.3%

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

      8. associate-+r+ 89.3%

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

      9. +-commutative 89.3%

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

      10. +-commutative 89.3%

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

   10. Applied egg-rr 89.3%

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

   11. Taylor expanded in beta around inf 90.2%

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

       1. mul-1-neg 90.2%

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

       2. +-commutative 90.2%

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

   13. Simplified 90.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3e+16)
   (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 6.0 (* beta (+ beta 5.0)))))
   (/
    (* (+ alpha 1.0) (/ (- 1.0 (/ (+ 3.0 (* alpha 2.0)) beta)) beta))
    (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+16) {
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / 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 <= 3d+16) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((alpha + 1.0d0) * ((1.0d0 - ((3.0d0 + (alpha * 2.0d0)) / beta)) / 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 <= 3e+16) {
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3e+16:
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 - Float64(Float64(3.0 + Float64(alpha * 2.0)) / beta)) / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3e+16)
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((alpha + 1.0) * ((1.0 - ((3.0 + (alpha * 2.0)) / beta)) / 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, 3e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(3.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e16

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in beta around 0 93.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   6. Taylor expanded in alpha around 0 61.9%

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

      1. +-commutative 61.9%

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

   8. Simplified 61.9%

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

   if 3e16 < beta

   1. Initial program 78.0%

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

   3. Simplified 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. associate-/l* 81.1%

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

      3. +-commutative 81.1%

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

      4. associate-+r+ 81.1%

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

      5. associate-*r* 81.1%

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

      6. pow2 81.1%

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

      7. associate-+r+ 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. *-lft-identity 81.1%

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

      2. +-commutative 81.1%

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

      3. associate-/r* 90.7%

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

      4. associate-+r+ 90.7%

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

      5. +-commutative 90.7%

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

      6. +-commutative 90.7%

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

      7. associate-+r+ 90.7%

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

      8. +-commutative 90.7%

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

      9. +-commutative 90.7%

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

   8. Simplified 90.7%

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

      1. associate-*r/ 90.6%

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

      2. div-inv 90.6%

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

      3. associate-+r+ 90.6%

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

      4. +-commutative 90.6%

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

      5. +-commutative 90.6%

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

      6. pow-flip 92.5%

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

      7. metadata-eval 92.5%

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

      8. associate-+r+ 92.5%

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

      9. +-commutative 92.5%

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

      10. +-commutative 92.5%

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

   10. Applied egg-rr 92.5%

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

   11. Taylor expanded in beta around inf 84.1%

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

       1. mul-1-neg 84.1%

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

       2. +-commutative 84.1%

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

   13. Simplified 84.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1 - \frac{3 + \alpha \cdot 2}{\beta}}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.4e+16)
   (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 6.0 (* beta (+ beta 5.0)))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.4e+16) {
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.4d+16) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.4e+16) {
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.4e+16:
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.4e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.4e+16)
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.4e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.4e16

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in beta around 0 93.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   6. Taylor expanded in alpha around 0 61.9%

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

      1. +-commutative 61.9%

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

   8. Simplified 61.9%

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

   if 1.4e16 < beta

   1. Initial program 78.0%

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

   3. Simplified 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. associate-/l* 81.1%

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

      3. +-commutative 81.1%

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

      4. associate-+r+ 81.1%

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

      5. associate-*r* 81.1%

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

      6. pow2 81.1%

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

      7. associate-+r+ 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. *-lft-identity 81.1%

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

      2. +-commutative 81.1%

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

      3. associate-/r* 90.7%

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

      4. associate-+r+ 90.7%

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

      5. +-commutative 90.7%

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

      6. +-commutative 90.7%

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

      7. associate-+r+ 90.7%

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

      8. +-commutative 90.7%

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

      9. +-commutative 90.7%

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

   8. Simplified 90.7%

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

      1. associate-*r/ 90.6%

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

      2. div-inv 90.6%

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

      3. associate-+r+ 90.6%

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

      4. +-commutative 90.6%

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

      5. +-commutative 90.6%

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

      6. pow-flip 92.5%

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

      7. metadata-eval 92.5%

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

      8. associate-+r+ 92.5%

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

      9. +-commutative 92.5%

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

      10. +-commutative 92.5%

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

   10. Applied egg-rr 92.5%

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

   11. Taylor expanded in beta around inf 83.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   0.08333333333333333
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], 0.08333333333333333, N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.8%

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

      1. associate-*r/ 60.8%

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

   8. Simplified 60.8%

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

   9. Taylor expanded in beta around 0 60.9%

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

   if 2.2999999999999998 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-/l* 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-*r* 81.5%

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

      6. pow2 81.5%

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

      7. associate-+r+ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-lft-identity 81.5%

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

      2. +-commutative 81.5%

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

      3. associate-/r* 90.3%

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

      4. associate-+r+ 90.3%

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

      5. +-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+r+ 90.3%

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

      8. +-commutative 90.3%

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

      9. +-commutative 90.3%

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

   8. Simplified 90.3%

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

      1. associate-*r/ 91.4%

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

      2. div-inv 91.3%

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

      3. associate-+r+ 91.3%

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

      4. +-commutative 91.3%

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

      5. +-commutative 91.3%

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

      6. pow-flip 93.0%

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

      7. metadata-eval 93.0%

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

      8. associate-+r+ 93.0%

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

      9. +-commutative 93.0%

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

      10. +-commutative 93.0%

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

   10. Applied egg-rr 93.0%

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

   11. Taylor expanded in beta around inf 79.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9)
   0.08333333333333333
   (* (+ alpha 1.0) (/ (/ 1.0 beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (alpha + 1.0) * ((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 <= 3.9d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = (alpha + 1.0d0) * ((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 <= 3.9) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (alpha + 1.0) * ((1.0 / beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.9:
		tmp = 0.08333333333333333
	else:
		tmp = (alpha + 1.0) * ((1.0 / beta) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.9)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(1.0 / beta) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.9)
		tmp = 0.08333333333333333;
	else
		tmp = (alpha + 1.0) * ((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, 3.9], 0.08333333333333333, N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999991

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.8%

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

      1. associate-*r/ 60.8%

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

   8. Simplified 60.8%

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

   9. Taylor expanded in beta around 0 60.9%

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

   if 3.89999999999999991 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-/l* 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-*r* 81.5%

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

      6. pow2 81.5%

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

      7. associate-+r+ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-lft-identity 81.5%

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

      2. +-commutative 81.5%

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

      3. associate-/r* 90.3%

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

      4. associate-+r+ 90.3%

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

      5. +-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+r+ 90.3%

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

      8. +-commutative 90.3%

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

      9. +-commutative 90.3%

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

   8. Simplified 90.3%

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

   9. Taylor expanded in beta around inf 84.4%

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

   10. Taylor expanded in beta around inf 79.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3) 0.08333333333333333 (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} 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 <= 2.3d0) then
        tmp = 0.08333333333333333d0
    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 <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333
	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 <= 2.3)
		tmp = 0.08333333333333333;
	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 <= 2.3)
		tmp = 0.08333333333333333;
	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, 2.3], 0.08333333333333333, N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.8%

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

      1. associate-*r/ 60.8%

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

   8. Simplified 60.8%

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

   9. Taylor expanded in beta around 0 60.9%

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

   if 2.2999999999999998 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-/l* 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-*r* 81.5%

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

      6. pow2 81.5%

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

      7. associate-+r+ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-lft-identity 81.5%

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

      2. +-commutative 81.5%

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

      3. associate-/r* 90.3%

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

      4. associate-+r+ 90.3%

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

      5. +-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+r+ 90.3%

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

      8. +-commutative 90.3%

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

      9. +-commutative 90.3%

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

   8. Simplified 90.3%

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

   9. Taylor expanded in beta around inf 84.4%

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

   10. Taylor expanded in alpha around 0 72.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3) 0.08333333333333333 (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} 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 <= 2.3d0) then
        tmp = 0.08333333333333333d0
    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 <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333
	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 <= 2.3)
		tmp = 0.08333333333333333;
	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 <= 2.3)
		tmp = 0.08333333333333333;
	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, 2.3], 0.08333333333333333, N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.8%

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

      1. associate-*r/ 60.8%

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

   8. Simplified 60.8%

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

   9. Taylor expanded in beta around 0 60.9%

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

   if 2.2999999999999998 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-/l* 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-*r* 81.5%

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

      6. pow2 81.5%

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

      7. associate-+r+ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-lft-identity 81.5%

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

      2. +-commutative 81.5%

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

      3. associate-/r* 90.3%

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

      4. associate-+r+ 90.3%

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

      5. +-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+r+ 90.3%

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

      8. +-commutative 90.3%

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

      9. +-commutative 90.3%

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

   8. Simplified 90.3%

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

   9. Taylor expanded in beta around inf 84.4%

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

   10. Taylor expanded in alpha around 0 72.4%

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

       1. associate-/r* 73.0%

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

       2. +-commutative 73.0%

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

   12. Simplified 73.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0) 0.08333333333333333 (/ 1.0 (* beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / (beta * 3.0)
	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(1.0 / Float64(beta * 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / (beta * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.0], 0.08333333333333333, N[(1.0 / N[(beta * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.8%

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

      1. associate-*r/ 60.8%

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

   8. Simplified 60.8%

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

   9. Taylor expanded in beta around 0 60.9%

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

   if 4 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-/l* 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-*r* 81.5%

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

      6. pow2 81.5%

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

      7. associate-+r+ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-lft-identity 81.5%

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

      2. +-commutative 81.5%

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

      3. associate-/r* 90.3%

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

      4. associate-+r+ 90.3%

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

      5. +-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+r+ 90.3%

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

      8. +-commutative 90.3%

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

      9. +-commutative 90.3%

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

   8. Simplified 90.3%

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

   9. Taylor expanded in beta around inf 84.4%

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

   10. Taylor expanded in alpha around 0 72.4%

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

   11. Taylor expanded in beta around 0 7.0%

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

       1. *-commutative 7.0%

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

   13. Simplified 7.0%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.9% 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 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.8%

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

      1. associate-*r/ 60.8%

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

   8. Simplified 60.8%

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

   9. Taylor expanded in beta around 0 60.9%

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

   if 4 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-/l* 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+r+ 81.6%

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

      5. associate-*r* 81.5%

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

      6. pow2 81.5%

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

      7. associate-+r+ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-lft-identity 81.5%

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

      2. +-commutative 81.5%

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

      3. associate-/r* 90.3%

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

      4. associate-+r+ 90.3%

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

      5. +-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+r+ 90.3%

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

      8. +-commutative 90.3%

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

      9. +-commutative 90.3%

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

   8. Simplified 90.3%

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

   9. Taylor expanded in beta around inf 84.4%

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

   10. Taylor expanded in alpha around 0 72.4%

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

   11. Taylor expanded in beta around 0 7.0%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.2% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 85.1%

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

5. Taylor expanded in beta around 0 65.1%

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

6. Taylor expanded in alpha around 0 42.3%

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

   1. associate-*r/ 42.3%

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

8. Simplified 42.3%

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

9. Taylor expanded in beta around 0 42.3%

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

10. Final simplification 42.3%

    \[\leadsto 0.08333333333333333 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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)))