Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.7%

Time: 18.0s

Alternatives: 17

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 3\right)}{\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{t\_0}} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{t\_1}{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}}}{1 + t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (+ (+ alpha beta) 2.0)))
   (if (<= beta 5.5e+153)
     (/
      1.0
      (*
       (/ (+ alpha (+ beta 3.0)) (/ (* (+ 1.0 alpha) (+ 1.0 beta)) t_0))
       t_0))
     (/
      (/ 1.0 (/ t_1 (* (+ 1.0 alpha) (+ 1.0 (/ (- -1.0 alpha) beta)))))
      (+ 1.0 t_1)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 5.5e+153) {
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0);
	} else {
		tmp = (1.0 / (t_1 / ((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))))) / (1.0 + t_1);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (alpha + beta) + 2.0d0
    if (beta <= 5.5d+153) then
        tmp = 1.0d0 / (((alpha + (beta + 3.0d0)) / (((1.0d0 + alpha) * (1.0d0 + beta)) / t_0)) * t_0)
    else
        tmp = (1.0d0 / (t_1 / ((1.0d0 + alpha) * (1.0d0 + (((-1.0d0) - alpha) / beta))))) / (1.0d0 + t_1)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 5.5e+153) {
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0);
	} else {
		tmp = (1.0 / (t_1 / ((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))))) / (1.0 + t_1);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 5.5e+153:
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0)
	else:
		tmp = (1.0 / (t_1 / ((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))))) / (1.0 + t_1)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 5.5e+153)
		tmp = Float64(1.0 / Float64(Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + beta)) / t_0)) * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(t_1 / Float64(Float64(1.0 + alpha) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))))) / Float64(1.0 + t_1));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 5.5e+153)
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0);
	else
		tmp = (1.0 / (t_1 / ((1.0 + alpha) * (1.0 + ((-1.0 - alpha) / beta))))) / (1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5.5e+153], N[(1.0 / N[(N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$1 / N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5000000000000003e153

   1. Initial program 98.4%

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

      1. associate-/l/ 98.0%

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

      2. +-commutative 98.0%

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

      3. associate-+l+ 98.0%

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

      4. *-commutative 98.0%

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

      5. metadata-eval 98.0%

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

      6. associate-+l+ 98.0%

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

      7. metadata-eval 98.0%

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

      8. +-commutative 98.0%

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

      9. +-commutative 98.0%

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

      10. +-commutative 98.0%

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

      11. metadata-eval 98.0%

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

      12. metadata-eval 98.0%

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

      13. associate-+l+ 98.0%

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

   3. Simplified 98.0%

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

      1. clear-num 98.0%

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

      2. inv-pow 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-+r+ 98.0%

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

      5. +-commutative 98.0%

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

      6. distribute-rgt1-in 98.0%

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

      7. fma-define 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. unpow-1 98.0%

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

      2. associate-/l* 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

      8. +-commutative 98.5%

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

      9. +-commutative 98.5%

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

      10. fma-undefine 98.5%

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

      11. +-commutative 98.5%

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

      12. *-commutative 98.5%

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

      13. +-commutative 98.5%

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

      14. associate-+r+ 98.5%

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

      15. distribute-lft1-in 98.5%

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

      16. +-commutative 98.5%

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

      17. +-commutative 98.5%

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

      18. +-commutative 98.5%

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

      19. +-commutative 98.5%

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

   8. Simplified 98.5%

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

   if 5.5000000000000003e153 < beta

   1. Initial program 57.8%

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

      1. clear-num 57.8%

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

      2. inv-pow 57.8%

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

      3. metadata-eval 57.8%

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

      4. associate-+r+ 57.8%

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

      5. +-commutative 57.8%

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

      6. *-commutative 57.8%

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

      7. associate-+r+ 57.8%

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

      8. +-commutative 57.8%

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

      9. distribute-rgt1-in 57.8%

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

      10. fma-define 57.8%

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

      11. metadata-eval 57.8%

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

      12. associate-+r+ 57.8%

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

   4. Applied egg-rr 57.8%

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

      1. unpow-1 57.8%

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

      2. associate-/r/ 57.7%

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

      3. +-commutative 57.7%

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

      4. +-commutative 57.7%

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

      5. +-commutative 57.7%

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

      6. +-commutative 57.7%

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

      7. fma-undefine 57.7%

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

      8. +-commutative 57.7%

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

      9. *-commutative 57.7%

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

      10. +-commutative 57.7%

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

      11. associate-+r+ 57.7%

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

      12. distribute-lft1-in 57.7%

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

      13. +-commutative 57.7%

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

      14. +-commutative 57.7%

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

      15. +-commutative 57.7%

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

      16. +-commutative 57.7%

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

      17. +-commutative 57.7%

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

   6. Simplified 57.7%

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

      1. clear-num 57.8%

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

      2. inv-pow 57.8%

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

      3. associate-+l+ 57.8%

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

   8. Applied egg-rr 57.8%

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

      1. unpow-1 57.8%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

       3. associate-+l+ 100.0%

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

       4. +-commutative 100.0%

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

       5. +-commutative 100.0%

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

       6. *-commutative 100.0%

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

       7. +-commutative 100.0%

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

       8. +-commutative 100.0%

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

       9. +-commutative 100.0%

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

       10. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. associate-+r+ 100.0%

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

       2. +-commutative 100.0%

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

       3. associate-*l/ 57.8%

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

       4. associate-/l* 100.0%

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

       5. +-commutative 100.0%

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

       6. associate-+r+ 100.0%

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

       7. +-commutative 100.0%

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

   14. Simplified 100.0%

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

   15. Taylor expanded in beta around inf 88.2%

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

       1. associate-*r/ 88.2%

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

       2. mul-1-neg 88.2%

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

   17. Simplified 88.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 3\right)}{\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{\alpha + \left(\beta + 2\right)}} \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 3\right)}{\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{t\_0}} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{1 + \alpha} \cdot t\_0}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.15e+154)
     (/
      1.0
      (*
       (/ (+ alpha (+ beta 3.0)) (/ (* (+ 1.0 alpha) (+ 1.0 beta)) t_0))
       t_0))
     (/
      (/ 1.0 (* (/ 1.0 (+ 1.0 alpha)) t_0))
      (+ 1.0 (+ (+ alpha beta) 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.15e+154) {
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0);
	} else {
		tmp = (1.0 / ((1.0 / (1.0 + alpha)) * t_0)) / (1.0 + ((alpha + beta) + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.15d+154) then
        tmp = 1.0d0 / (((alpha + (beta + 3.0d0)) / (((1.0d0 + alpha) * (1.0d0 + beta)) / t_0)) * t_0)
    else
        tmp = (1.0d0 / ((1.0d0 / (1.0d0 + alpha)) * t_0)) / (1.0d0 + ((alpha + beta) + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.15e+154) {
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0);
	} else {
		tmp = (1.0 / ((1.0 / (1.0 + alpha)) * t_0)) / (1.0 + ((alpha + beta) + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.15e+154:
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0)
	else:
		tmp = (1.0 / ((1.0 / (1.0 + alpha)) * t_0)) / (1.0 + ((alpha + beta) + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.15e+154)
		tmp = Float64(1.0 / Float64(Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + beta)) / t_0)) * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 / Float64(1.0 + alpha)) * t_0)) / Float64(1.0 + Float64(Float64(alpha + beta) + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.15e+154)
		tmp = 1.0 / (((alpha + (beta + 3.0)) / (((1.0 + alpha) * (1.0 + beta)) / t_0)) * t_0);
	else
		tmp = (1.0 / ((1.0 / (1.0 + alpha)) * t_0)) / (1.0 + ((alpha + beta) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.15e+154], N[(1.0 / N[(N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(1.0 / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15e154

   1. Initial program 98.4%

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

      1. associate-/l/ 98.0%

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

      2. +-commutative 98.0%

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

      3. associate-+l+ 98.0%

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

      4. *-commutative 98.0%

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

      5. metadata-eval 98.0%

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

      6. associate-+l+ 98.0%

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

      7. metadata-eval 98.0%

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

      8. +-commutative 98.0%

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

      9. +-commutative 98.0%

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

      10. +-commutative 98.0%

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

      11. metadata-eval 98.0%

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

      12. metadata-eval 98.0%

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

      13. associate-+l+ 98.0%

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

   3. Simplified 98.0%

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

      1. clear-num 98.0%

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

      2. inv-pow 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-+r+ 98.0%

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

      5. +-commutative 98.0%

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

      6. distribute-rgt1-in 98.0%

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

      7. fma-define 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. unpow-1 98.0%

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

      2. associate-/l* 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

      8. +-commutative 98.5%

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

      9. +-commutative 98.5%

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

      10. fma-undefine 98.5%

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

      11. +-commutative 98.5%

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

      12. *-commutative 98.5%

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

      13. +-commutative 98.5%

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

      14. associate-+r+ 98.5%

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

      15. distribute-lft1-in 98.5%

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

      16. +-commutative 98.5%

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

      17. +-commutative 98.5%

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

      18. +-commutative 98.5%

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

      19. +-commutative 98.5%

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

   8. Simplified 98.5%

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

   if 1.15e154 < beta

   1. Initial program 57.8%

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

      1. clear-num 57.8%

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

      2. inv-pow 57.8%

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

      3. metadata-eval 57.8%

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

      4. associate-+r+ 57.8%

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

      5. +-commutative 57.8%

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

      6. *-commutative 57.8%

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

      7. associate-+r+ 57.8%

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

      8. +-commutative 57.8%

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

      9. distribute-rgt1-in 57.8%

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

      10. fma-define 57.8%

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

      11. metadata-eval 57.8%

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

      12. associate-+r+ 57.8%

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

   4. Applied egg-rr 57.8%

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

      1. unpow-1 57.8%

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

      2. associate-/r/ 57.7%

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

      3. +-commutative 57.7%

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

      4. +-commutative 57.7%

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

      5. +-commutative 57.7%

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

      6. +-commutative 57.7%

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

      7. fma-undefine 57.7%

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

      8. +-commutative 57.7%

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

      9. *-commutative 57.7%

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

      10. +-commutative 57.7%

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

      11. associate-+r+ 57.7%

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

      12. distribute-lft1-in 57.7%

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

      13. +-commutative 57.7%

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

      14. +-commutative 57.7%

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

      15. +-commutative 57.7%

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

      16. +-commutative 57.7%

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

      17. +-commutative 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in beta around inf 87.9%

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

      1. +-commutative 87.9%

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

   9. Simplified 87.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 3\right)}{\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{\alpha + \left(\beta + 2\right)}} \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{1 + \alpha} \cdot \left(\alpha + \left(\beta + 2\right)\right)}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{t\_1 \cdot \left(t\_1 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{t\_0}{1 + \alpha}}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+99)
     (/ (* (+ 1.0 alpha) (+ 1.0 beta)) (* t_1 (* t_1 (+ alpha (+ beta 3.0)))))
     (/ (/ 1.0 (/ t_0 (+ 1.0 alpha))) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+99) {
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_1 * (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 5d+99) then
        tmp = ((1.0d0 + alpha) * (1.0d0 + beta)) / (t_1 * (t_1 * (alpha + (beta + 3.0d0))))
    else
        tmp = (1.0d0 / (t_0 / (1.0d0 + alpha))) / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+99) {
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_1 * (t_1 * (alpha + (beta + 3.0))));
	} else {
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+99:
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_1 * (t_1 * (alpha + (beta + 3.0))))
	else:
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+99)
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + beta)) / Float64(t_1 * Float64(t_1 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_0 / Float64(1.0 + alpha))) / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+99)
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_1 * (t_1 * (alpha + (beta + 3.0))));
	else
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+99], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$0 / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000008e99

   1. Initial program 98.8%

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

   3. Simplified 93.6%

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

   if 5.00000000000000008e99 < beta

   1. Initial program 69.6%

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

      1. clear-num 69.7%

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

      2. inv-pow 69.7%

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

      3. metadata-eval 69.7%

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

      4. associate-+r+ 69.7%

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

      5. +-commutative 69.7%

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

      6. *-commutative 69.7%

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

      7. associate-+r+ 69.7%

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

      8. +-commutative 69.7%

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

      9. distribute-rgt1-in 69.7%

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

      10. fma-define 69.7%

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

      11. metadata-eval 69.7%

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

      12. associate-+r+ 69.7%

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

   4. Applied egg-rr 69.7%

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

      1. unpow-1 69.7%

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

      2. associate-/r/ 69.6%

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

      3. +-commutative 69.6%

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

      4. +-commutative 69.6%

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

      5. +-commutative 69.6%

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

      6. +-commutative 69.6%

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

      7. fma-undefine 69.6%

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

      8. +-commutative 69.6%

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

      9. *-commutative 69.6%

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

      10. +-commutative 69.6%

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

      11. associate-+r+ 69.6%

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

      12. distribute-lft1-in 69.6%

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

      13. +-commutative 69.6%

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

      14. +-commutative 69.6%

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

      15. +-commutative 69.6%

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

      16. +-commutative 69.6%

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

      17. +-commutative 69.6%

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

   6. Simplified 69.6%

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

      1. clear-num 69.7%

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

      2. inv-pow 69.7%

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

      3. associate-+l+ 69.7%

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

   8. Applied egg-rr 69.7%

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

      1. unpow-1 69.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. associate-*l/ 99.8%

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

       2. *-un-lft-identity 99.8%

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

       3. associate-+l+ 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

       6. *-commutative 99.8%

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

       7. +-commutative 99.8%

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

       8. +-commutative 99.8%

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

       9. +-commutative 99.8%

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

       10. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. associate-+r+ 99.8%

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

       2. +-commutative 99.8%

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

       3. associate-*l/ 69.7%

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

       4. associate-/l* 99.8%

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

       5. +-commutative 99.8%

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

       6. associate-+r+ 99.8%

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

       7. +-commutative 99.8%

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

   14. Simplified 99.8%

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

   15. Taylor expanded in beta around inf 88.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{1 + \alpha}}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. clear-num 91.7%

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

   2. inv-pow 91.7%

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

   3. metadata-eval 91.7%

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

   4. associate-+r+ 91.7%

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

   5. +-commutative 91.7%

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

   6. *-commutative 91.7%

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

   7. associate-+r+ 91.7%

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

   8. +-commutative 91.7%

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

   9. distribute-rgt1-in 91.7%

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

   10. fma-define 91.7%

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

   11. metadata-eval 91.7%

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

   12. associate-+r+ 91.7%

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

4. Applied egg-rr 91.7%

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

   1. unpow-1 91.7%

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

   2. associate-/r/ 91.7%

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

   3. +-commutative 91.7%

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

   4. +-commutative 91.7%

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

   5. +-commutative 91.7%

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

   6. +-commutative 91.7%

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

   7. fma-undefine 91.7%

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

   8. +-commutative 91.7%

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

   9. *-commutative 91.7%

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

   10. +-commutative 91.7%

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

   11. associate-+r+ 91.7%

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

   12. distribute-lft1-in 91.7%

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

   13. +-commutative 91.7%

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

   14. +-commutative 91.7%

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

   15. +-commutative 91.7%

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

   16. +-commutative 91.7%

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

   17. +-commutative 91.7%

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

6. Simplified 91.7%

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

   1. clear-num 91.7%

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

   2. inv-pow 91.7%

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

   3. associate-+l+ 91.7%

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

8. Applied egg-rr 91.7%

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

   1. unpow-1 91.7%

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

   2. associate-/l* 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

10. Simplified 99.7%

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

    1. associate-*l/ 99.8%

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

    2. *-un-lft-identity 99.8%

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

    3. associate-+l+ 99.8%

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

    4. +-commutative 99.8%

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

    5. +-commutative 99.8%

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

    6. *-commutative 99.8%

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

    7. +-commutative 99.8%

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

    8. +-commutative 99.8%

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

    9. +-commutative 99.8%

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

    10. +-commutative 99.8%

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

12. Applied egg-rr 99.8%

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

    1. associate-+r+ 99.8%

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

    2. +-commutative 99.8%

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

    3. associate-*l/ 91.7%

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

    4. associate-/l* 99.8%

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

    5. +-commutative 99.8%

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

    6. associate-+r+ 99.8%

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

    7. +-commutative 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 5: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 65:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + 2}{1 + \alpha}}}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 65.0)
   (/
    (/ 1.0 (* (+ alpha (+ beta 2.0)) (/ (+ alpha 2.0) (+ 1.0 alpha))))
    (+ alpha 3.0))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 65.0) {
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 65.0d0) then
        tmp = (1.0d0 / ((alpha + (beta + 2.0d0)) * ((alpha + 2.0d0) / (1.0d0 + alpha)))) / (alpha + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 65.0) {
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 65.0:
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 65.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(alpha + 2.0) / Float64(1.0 + alpha)))) / Float64(alpha + 3.0));
	else
		tmp = Float64(Float64(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 <= 65.0)
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 65.0], N[(N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 65

   1. Initial program 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-rgt1-in 99.8%

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

      10. fma-define 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-undefine 99.8%

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

      8. +-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

      12. distribute-lft1-in 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around 0 97.9%

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

      1. +-commutative 97.9%

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

   9. Simplified 97.9%

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

   10. Taylor expanded in beta around 0 97.4%

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

       1. +-commutative 97.4%

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

   12. Simplified 97.4%

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

   if 65 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf 81.9%

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

   4. Taylor expanded in alpha around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. associate-+r+ 81.9%

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

      3. +-commutative 81.9%

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

   6. Simplified 81.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 65:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + 2}{1 + \alpha}}}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 18:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + 2}{1 + \alpha}}}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{t\_0}{1 + \alpha}}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 18.0)
     (/
      (/ 1.0 (* (+ alpha (+ beta 2.0)) (/ (+ alpha 2.0) (+ 1.0 alpha))))
      (+ alpha 3.0))
     (/ (/ 1.0 (/ t_0 (+ 1.0 alpha))) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 18.0) {
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0);
	} else {
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 18.0d0) then
        tmp = (1.0d0 / ((alpha + (beta + 2.0d0)) * ((alpha + 2.0d0) / (1.0d0 + alpha)))) / (alpha + 3.0d0)
    else
        tmp = (1.0d0 / (t_0 / (1.0d0 + alpha))) / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 18.0) {
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0);
	} else {
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 18.0:
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0)
	else:
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 18.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(alpha + 2.0) / Float64(1.0 + alpha)))) / Float64(alpha + 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(t_0 / Float64(1.0 + alpha))) / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 18.0)
		tmp = (1.0 / ((alpha + (beta + 2.0)) * ((alpha + 2.0) / (1.0 + alpha)))) / (alpha + 3.0);
	else
		tmp = (1.0 / (t_0 / (1.0 + alpha))) / (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 18.0], N[(N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$0 / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 18

   1. Initial program 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-rgt1-in 99.8%

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

      10. fma-define 99.8%

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

      11. metadata-eval 99.8%

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

      12. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-undefine 99.8%

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

      8. +-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

      12. distribute-lft1-in 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

      17. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in beta around 0 97.9%

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

      1. +-commutative 97.9%

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

   9. Simplified 97.9%

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

   10. Taylor expanded in beta around 0 97.4%

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

       1. +-commutative 97.4%

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

   12. Simplified 97.4%

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

   if 18 < beta

   1. Initial program 76.3%

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

      1. clear-num 76.2%

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

      2. inv-pow 76.2%

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

      3. metadata-eval 76.2%

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

      4. associate-+r+ 76.2%

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

      5. +-commutative 76.2%

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

      6. *-commutative 76.2%

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

      7. associate-+r+ 76.2%

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

      8. +-commutative 76.2%

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

      9. distribute-rgt1-in 76.3%

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

      10. fma-define 76.3%

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

      11. metadata-eval 76.3%

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

      12. associate-+r+ 76.3%

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

   4. Applied egg-rr 76.3%

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

      1. unpow-1 76.3%

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

      2. associate-/r/ 76.2%

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

      3. +-commutative 76.2%

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

      4. +-commutative 76.2%

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

      5. +-commutative 76.2%

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

      6. +-commutative 76.2%

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

      7. fma-undefine 76.2%

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

      8. +-commutative 76.2%

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

      9. *-commutative 76.2%

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

      10. +-commutative 76.2%

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

      11. associate-+r+ 76.2%

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

      12. distribute-lft1-in 76.2%

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

      13. +-commutative 76.2%

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

      14. +-commutative 76.2%

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

      15. +-commutative 76.2%

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

      16. +-commutative 76.2%

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

      17. +-commutative 76.2%

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

   6. Simplified 76.2%

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

      1. clear-num 76.2%

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

      2. inv-pow 76.2%

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

      3. associate-+l+ 76.2%

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

   8. Applied egg-rr 76.2%

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

      1. unpow-1 76.2%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. associate-*l/ 99.8%

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

       2. *-un-lft-identity 99.8%

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

       3. associate-+l+ 99.8%

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

       4. +-commutative 99.8%

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

       5. +-commutative 99.8%

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

       6. *-commutative 99.8%

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

       7. +-commutative 99.8%

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

       8. +-commutative 99.8%

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

       9. +-commutative 99.8%

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

       10. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. associate-+r+ 99.8%

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

       2. +-commutative 99.8%

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

       3. associate-*l/ 76.3%

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

       4. associate-/l* 99.8%

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

       5. +-commutative 99.8%

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

       6. associate-+r+ 99.8%

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

       7. +-commutative 99.8%

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

   14. Simplified 99.8%

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

   15. Taylor expanded in beta around inf 82.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 18:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + 2}{1 + \alpha}}}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{1 + \alpha}}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.7e+30)
   (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.7e+30) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.7d+30) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.7e+30) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.7e+30:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.7e+30)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(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 <= 4.7e+30)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.7e+30], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6999999999999999e30

   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 94.5%

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

   5. Taylor expanded in alpha around 0 82.9%

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

   6. Taylor expanded in alpha around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. +-commutative 64.6%

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

   8. Simplified 64.6%

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

   if 4.6999999999999999e30 < beta

   1. Initial program 74.8%

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

   3. Taylor expanded in beta around inf 84.3%

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

   4. Taylor expanded in alpha around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. associate-+r+ 84.3%

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

      3. +-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8e+30)
   (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (+ 6.0 (* beta (+ beta 5.0)))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+30) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.8d+30) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+30) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8e+30:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8e+30)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(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 <= 4.8e+30)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8e+30], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.7999999999999999e30

   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 94.5%

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

   5. Taylor expanded in alpha around 0 82.9%

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

   6. Taylor expanded in alpha around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. +-commutative 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in beta around 0 64.6%

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

       1. +-commutative 64.6%

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

   11. Simplified 64.6%

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

   if 4.7999999999999999e30 < beta

   1. Initial program 74.8%

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

   3. Taylor expanded in beta around inf 84.3%

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

   4. Taylor expanded in alpha around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. associate-+r+ 84.3%

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

      3. +-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (+ 6.0 (* beta 5.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.8d0) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * (6.0d0 + (beta * 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * 5.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(6.0 + Float64(beta * 5.0))));
	else
		tmp = Float64(Float64(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 <= 4.8)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * (6.0 + (beta * 5.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   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 94.4%

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

   5. Taylor expanded in alpha around 0 82.3%

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

   6. Taylor expanded in alpha around 0 65.2%

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

      1. +-commutative 65.2%

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

      2. +-commutative 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in beta around 0 64.1%

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

       1. *-commutative 64.1%

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

   11. Simplified 64.1%

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

   if 4.79999999999999982 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf 81.9%

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

   4. Taylor expanded in alpha around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. associate-+r+ 81.9%

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

      3. +-commutative 81.9%

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

   6. Simplified 81.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.16e+35)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.16e+35) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.16d+35) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.16e+35) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.16e+35:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.16e+35)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / 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 <= 1.16e+35)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.16e+35], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1600000000000001e35

   1. Initial program 99.8%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 84.7%

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

      1. +-commutative 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in alpha around 0 62.9%

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

   if 1.1600000000000001e35 < beta

   1. Initial program 74.8%

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

   3. Taylor expanded in beta around inf 84.3%

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

   4. Taylor expanded in alpha around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. associate-+r+ 84.3%

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

      3. +-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (+ 6.0 (* beta 5.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.5d0) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * (6.0d0 + (beta * 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(6.0 + Float64(beta * 5.0))));
	else
		tmp = Float64(Float64(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 <= 4.5)
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in alpha around 0 82.3%

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

   6. Taylor expanded in alpha around 0 65.2%

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

      1. +-commutative 65.2%

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

      2. +-commutative 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in beta around 0 64.1%

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

       1. *-commutative 64.1%

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

   11. Simplified 64.1%

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

   12. Taylor expanded in alpha around 0 62.5%

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

   if 4.5 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf 81.9%

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

   4. Taylor expanded in alpha around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. associate-+r+ 81.9%

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

      3. +-commutative 81.9%

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

   6. Simplified 81.9%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(Float64(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 <= 2.1)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.1], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   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 94.3%

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

   5. Taylor expanded in alpha around 0 82.8%

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

   6. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. +-commutative 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in beta around 0 63.2%

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

       1. +-commutative 63.2%

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

   11. Simplified 63.2%

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

   12. Taylor expanded in alpha around 0 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 61.3%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.10000000000000009 < beta

   1. Initial program 76.5%

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

   3. Taylor expanded in beta around inf 81.0%

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

   4. Taylor expanded in alpha around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-+r+ 81.0%

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

      3. +-commutative 81.0%

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

   6. Simplified 81.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} 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 <= 2.2d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    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 <= 2.2) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	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 <= 2.2)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	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 <= 2.2)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	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, 2.2], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   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 94.3%

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

   5. Taylor expanded in alpha around 0 82.8%

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

   6. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. +-commutative 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in beta around 0 63.2%

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

       1. +-commutative 63.2%

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

   11. Simplified 63.2%

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

   12. Taylor expanded in alpha around 0 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 61.3%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.2000000000000002 < beta

   1. Initial program 76.5%

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

   3. Taylor expanded in beta around inf 81.0%

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

   4. Taylor expanded in alpha around 0 73.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.95)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.95d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.95:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.95)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.95)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.95], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9500000000000002

   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 94.3%

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

   5. Taylor expanded in alpha around 0 82.8%

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

   6. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. +-commutative 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in beta around 0 63.2%

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

       1. +-commutative 63.2%

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

   11. Simplified 63.2%

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

   12. Taylor expanded in alpha around 0 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 61.3%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 2.9500000000000002 < beta

   1. Initial program 76.5%

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

   3. Taylor expanded in beta around inf 81.0%

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

   4. Taylor expanded in beta around inf 80.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.95)
   (+ 0.08333333333333333 (* alpha -0.041666666666666664))
   (/ 0.2 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.95d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.041666666666666664d0))
    else
        tmp = 0.2d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95) {
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.95:
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664)
	else:
		tmp = 0.2 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.95)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.041666666666666664));
	else
		tmp = Float64(0.2 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.95)
		tmp = 0.08333333333333333 + (alpha * -0.041666666666666664);
	else
		tmp = 0.2 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.95], N[(0.08333333333333333 + N[(alpha * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(0.2 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.94999999999999996

   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 94.3%

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

   5. Taylor expanded in alpha around 0 82.8%

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

   6. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. +-commutative 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in beta around 0 63.2%

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

       1. +-commutative 63.2%

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

   11. Simplified 63.2%

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

   12. Taylor expanded in alpha around 0 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + -0.041666666666666664 \cdot \alpha} \]
   13. Step-by-step derivation

       1. *-commutative 61.3%

          \[\leadsto 0.08333333333333333 + \color{blue}{\alpha \cdot -0.041666666666666664} \]

   14. Simplified 61.3%

       \[\leadsto \color{blue}{0.08333333333333333 + \alpha \cdot -0.041666666666666664} \]

   if 1.94999999999999996 < beta

   1. Initial program 76.5%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in alpha around 0 69.2%

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

   6. Taylor expanded in alpha around 0 66.1%

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

      1. +-commutative 66.1%

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

      2. +-commutative 66.1%

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

   8. Simplified 66.1%

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

   9. Taylor expanded in beta around 0 46.9%

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

       1. *-commutative 46.9%

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

   11. Simplified 46.9%

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

   12. Taylor expanded in beta around inf 6.4%

       \[\leadsto \color{blue}{\frac{0.2}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4) 0.08333333333333333 (/ 0.2 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.2d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = 0.08333333333333333
	else:
		tmp = 0.2 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.2 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = 0.08333333333333333;
	else
		tmp = 0.2 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.4], 0.08333333333333333, N[(0.2 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   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 94.3%

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

   5. Taylor expanded in alpha around 0 82.8%

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

   6. Taylor expanded in alpha around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. +-commutative 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in beta around 0 63.2%

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

       1. +-commutative 63.2%

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

   11. Simplified 63.2%

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

   12. Taylor expanded in alpha around 0 62.0%

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

   if 2.39999999999999991 < beta

   1. Initial program 76.5%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in alpha around 0 69.2%

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

   6. Taylor expanded in alpha around 0 66.1%

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

      1. +-commutative 66.1%

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

      2. +-commutative 66.1%

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

   8. Simplified 66.1%

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

   9. Taylor expanded in beta around 0 46.9%

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

       1. *-commutative 46.9%

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

   11. Simplified 46.9%

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

   12. Taylor expanded in beta around inf 6.4%

       \[\leadsto \color{blue}{\frac{0.2}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 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 91.7%

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

3. Simplified 80.6%

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

5. Taylor expanded in alpha around 0 78.1%

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

6. Taylor expanded in alpha around 0 65.8%

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

   1. +-commutative 65.8%

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

   2. +-commutative 65.8%

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

8. Simplified 65.8%

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

9. Taylor expanded in beta around 0 42.9%

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

    1. +-commutative 42.9%

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

11. Simplified 42.9%

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

12. Taylor expanded in alpha around 0 41.7%

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

13. Final simplification 41.7%

    \[\leadsto 0.08333333333333333 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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)))