Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.6%

Time: 15.1s

Alternatives: 22

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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+78}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\left(\alpha + \frac{1}{\beta}\right) + \frac{\alpha}{\beta}\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+78)
     (/
      1.0
      (*
       t_0
       (/ (+ alpha (+ beta 3.0)) (/ (* (+ beta 1.0) (+ alpha 1.0)) t_0))))
     (/
      (/
       (+
        (+ 1.0 (+ (+ alpha (/ 1.0 beta)) (/ alpha beta)))
        (* (/ (+ 4.0 (* alpha 2.0)) beta) (- -1.0 alpha)))
       beta)
      (+ 1.0 (+ 2.0 (+ alpha beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+78) {
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)));
	} else {
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1d+78) then
        tmp = 1.0d0 / (t_0 * ((alpha + (beta + 3.0d0)) / (((beta + 1.0d0) * (alpha + 1.0d0)) / t_0)))
    else
        tmp = (((1.0d0 + ((alpha + (1.0d0 / beta)) + (alpha / beta))) + (((4.0d0 + (alpha * 2.0d0)) / beta) * ((-1.0d0) - alpha))) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+78) {
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)));
	} else {
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1e+78:
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)))
	else:
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1e+78)
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / t_0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(alpha + Float64(1.0 / beta)) + Float64(alpha / beta))) + Float64(Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta) * Float64(-1.0 - alpha))) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1e+78)
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)));
	else
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+78], N[(1.0 / N[(t$95$0 * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(N[(alpha + N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000001e78

   1. Initial program 99.3%

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

      1. associate-/l/ 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+l+ 98.3%

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

      4. *-commutative 98.3%

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

      5. metadata-eval 98.3%

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

      6. associate-+l+ 98.3%

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

      7. metadata-eval 98.3%

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

      8. +-commutative 98.3%

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

      9. +-commutative 98.3%

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

      10. +-commutative 98.3%

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

      11. metadata-eval 98.3%

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

      12. metadata-eval 98.3%

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

      13. associate-+l+ 98.3%

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

   3. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-+r+ 98.3%

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

      5. +-commutative 98.3%

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

      6. distribute-rgt1-in 98.3%

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

      7. fma-define 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. unpow-1 98.3%

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

      2. associate-/l* 98.7%

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

      3. +-commutative 98.7%

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

      4. +-commutative 98.7%

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

      5. +-commutative 98.7%

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

      6. +-commutative 98.7%

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

      7. +-commutative 98.7%

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

      8. +-commutative 98.7%

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

      9. +-commutative 98.7%

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

      10. fma-undefine 98.7%

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

      11. +-commutative 98.7%

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

      12. *-commutative 98.7%

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

      13. +-commutative 98.7%

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

      14. associate-+r+ 98.7%

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

      15. distribute-lft1-in 98.7%

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

      16. +-commutative 98.7%

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

      17. +-commutative 98.7%

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

      18. +-commutative 98.7%

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

      19. +-commutative 98.7%

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

   8. Simplified 98.7%

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

   if 1.00000000000000001e78 < 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 78.0%

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

      1. associate-+r+ 78.0%

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

      2. associate-/l* 85.9%

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

   5. Simplified 85.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+78}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\left(\alpha + \frac{1}{\beta}\right) + \frac{\alpha}{\beta}\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

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

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))))
   (pow
    (sqrt
     (* (/ t_0 (+ beta 1.0)) (/ (+ alpha (+ beta 3.0)) (/ (+ alpha 1.0) t_0))))
    -2.0)))

C

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

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 = sqrt(((t_0 / (beta + 1.0d0)) * ((alpha + (beta + 3.0d0)) / ((alpha + 1.0d0) / t_0)))) ** (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. associate-/l/ 92.0%

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

   2. +-commutative 92.0%

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

   3. associate-+l+ 92.0%

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

   4. *-commutative 92.0%

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

   5. metadata-eval 92.0%

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

   6. associate-+l+ 92.0%

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

   7. metadata-eval 92.0%

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

   8. +-commutative 92.0%

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

   9. +-commutative 92.0%

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

   10. +-commutative 92.0%

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

   11. metadata-eval 92.0%

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

   12. metadata-eval 92.0%

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

   13. associate-+l+ 92.0%

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

3. Simplified 92.0%

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

   1. clear-num 92.0%

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

   2. inv-pow 92.0%

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

   3. *-commutative 92.0%

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

   4. associate-+r+ 92.0%

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

   5. +-commutative 92.0%

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

   6. distribute-rgt1-in 92.0%

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

   7. fma-define 92.0%

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

6. Applied egg-rr 92.0%

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

   1. unpow-1 92.0%

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

   2. associate-/l* 92.8%

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

   3. +-commutative 92.8%

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

   4. +-commutative 92.8%

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

   5. +-commutative 92.8%

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

   6. +-commutative 92.8%

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

   7. +-commutative 92.8%

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

   8. +-commutative 92.8%

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

   9. +-commutative 92.8%

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

   10. fma-undefine 92.8%

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

   11. +-commutative 92.8%

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

   12. *-commutative 92.8%

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

   13. +-commutative 92.8%

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

   14. associate-+r+ 92.8%

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

   15. distribute-lft1-in 92.8%

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

   16. +-commutative 92.8%

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

   17. +-commutative 92.8%

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

   18. +-commutative 92.8%

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

   19. +-commutative 92.8%

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

8. Simplified 92.8%

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

   1. inv-pow 92.8%

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

   2. add-sqr-sqrt 92.2%

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

   3. unpow-prod-down 91.8%

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

10. Applied egg-rr 95.1%

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

    1. pow-sqr 95.6%

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

    2. times-frac 98.4%

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

    3. metadata-eval 98.4%

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

12. Simplified 98.4%

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

13. Final simplification 98.4%

    \[\leadsto {\left(\sqrt{\frac{\alpha + \left(\beta + 2\right)}{\beta + 1} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}}\right)}^{-2} \]
14. Add Preprocessing

Alternative 3: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5.8e+97)
     (/
      1.0
      (*
       t_0
       (/ (+ alpha (+ beta 3.0)) (/ (* (+ beta 1.0) (+ alpha 1.0)) t_0))))
     (*
      (/ (+ alpha 1.0) t_0)
      (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5.8e+97) {
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5.8d+97) then
        tmp = 1.0d0 / (t_0 * ((alpha + (beta + 3.0d0)) / (((beta + 1.0d0) * (alpha + 1.0d0)) / t_0)))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5.8e+97) {
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5.8e+97:
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	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 <= 5.8e+97)
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / t_0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	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 <= 5.8e+97)
		tmp = 1.0 / (t_0 * ((alpha + (beta + 3.0)) / (((beta + 1.0) * (alpha + 1.0)) / t_0)));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.8e+97], N[(1.0 / N[(t$95$0 * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999974e97

   1. Initial program 97.8%

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

      1. associate-/l/ 96.9%

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

      2. +-commutative 96.9%

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

      3. associate-+l+ 96.9%

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

      4. *-commutative 96.9%

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

      5. metadata-eval 96.9%

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

      6. associate-+l+ 96.9%

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

      7. metadata-eval 96.9%

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

      8. +-commutative 96.9%

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

      9. +-commutative 96.9%

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

      10. +-commutative 96.9%

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

      11. metadata-eval 96.9%

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

      12. metadata-eval 96.9%

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

      13. associate-+l+ 96.9%

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

   3. Simplified 96.9%

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

      1. clear-num 96.9%

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

      2. inv-pow 96.9%

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

      3. *-commutative 96.9%

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

      4. associate-+r+ 96.9%

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

      5. +-commutative 96.9%

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

      6. distribute-rgt1-in 96.9%

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

      7. fma-define 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. unpow-1 96.9%

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

      2. associate-/l* 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. +-commutative 97.3%

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

      6. +-commutative 97.3%

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

      7. +-commutative 97.3%

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

      8. +-commutative 97.3%

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

      9. +-commutative 97.3%

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

      10. fma-undefine 97.3%

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

      11. +-commutative 97.3%

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

      12. *-commutative 97.3%

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

      13. +-commutative 97.3%

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

      14. associate-+r+ 97.3%

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

      15. distribute-lft1-in 97.3%

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

      16. +-commutative 97.3%

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

      17. +-commutative 97.3%

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

      18. +-commutative 97.3%

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

      19. +-commutative 97.3%

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

   8. Simplified 97.3%

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

   if 5.79999999999999974e97 < beta

   1. Initial program 78.4%

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

   3. Simplified 57.7%

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

      1. times-frac 86.1%

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

      2. +-commutative 86.1%

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

   6. Applied egg-rr 86.1%

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

   7. Taylor expanded in beta around inf 91.2%

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

      1. mul-1-neg 91.2%

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

   9. Simplified 91.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{t\_0 \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.3e+86)
     (/ (* (+ beta 1.0) (+ alpha 1.0)) (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
     (*
      (/ (+ alpha 1.0) t_0)
      (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.3e+86) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.3d+86) then
        tmp = ((beta + 1.0d0) * (alpha + 1.0d0)) / (t_0 * (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.3e+86) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.3e+86:
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	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 <= 2.3e+86)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	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 <= 2.3e+86)
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.3e+86], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999999e86

   1. Initial program 98.3%

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

   3. Simplified 92.2%

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

   if 2.2999999999999999e86 < beta

   1. Initial program 77.8%

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

   3. Simplified 58.1%

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

      1. times-frac 86.8%

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

      2. +-commutative 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in beta around inf 89.9%

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

      1. mul-1-neg 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha + 1}{t\_0}\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+73}:\\ \;\;\;\;t\_1 \cdot \frac{\beta + 1}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ (+ alpha 1.0) t_0)))
   (if (<= beta 2e+73)
     (* t_1 (/ (+ beta 1.0) (* t_0 (+ alpha (+ beta 3.0)))))
     (* t_1 (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + 1.0) / t_0;
	double tmp;
	if (beta <= 2e+73) {
		tmp = t_1 * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = t_1 * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (alpha + 1.0d0) / t_0
    if (beta <= 2d+73) then
        tmp = t_1 * ((beta + 1.0d0) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = t_1 * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + 1.0) / t_0;
	double tmp;
	if (beta <= 2e+73) {
		tmp = t_1 * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = t_1 * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (alpha + 1.0) / t_0
	tmp = 0
	if beta <= 2e+73:
		tmp = t_1 * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = t_1 * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	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 + 1.0) / t_0)
	tmp = 0.0
	if (beta <= 2e+73)
		tmp = Float64(t_1 * Float64(Float64(beta + 1.0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	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 + 1.0) / t_0;
	tmp = 0.0;
	if (beta <= 2e+73)
		tmp = t_1 * ((beta + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = t_1 * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2e+73], N[(t$95$1 * N[(N[(beta + 1.0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999997e73

   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 94.0%

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

      1. times-frac 98.8%

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

      2. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   if 1.99999999999999997e73 < beta

   1. Initial program 75.3%

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

   3. Simplified 55.9%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

   7. Taylor expanded in beta around inf 84.7%

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

      1. mul-1-neg 84.7%

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

   9. Simplified 84.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2100000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2100000000.0)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 2.0) (+ 3.0 (+ alpha beta))))
   (*
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
    (/ (- 1.0 (/ (+ 4.0 (* alpha 2.0)) beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2100000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2100000000.0d0) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * ((1.0d0 - ((4.0d0 + (alpha * 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2100000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2100000000.0:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2100000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(Float64(4.0 + Float64(alpha * 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2100000000.0)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - ((4.0 + (alpha * 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2100000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(4.0 + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.1e9

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

      1. +-commutative 81.0%

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

   7. Simplified 81.0%

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

   8. Taylor expanded in alpha around 0 66.0%

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

   if 2.1e9 < beta

   1. Initial program 80.8%

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

   3. Simplified 63.7%

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

      1. times-frac 89.5%

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

      2. +-commutative 89.5%

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

   6. Applied egg-rr 89.5%

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

   7. Taylor expanded in beta around inf 81.8%

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

      1. mul-1-neg 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2100000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.6e+15)
   (/ (+ beta 1.0) (* (+ beta 2.0) (* (+ beta 2.0) (+ 3.0 (+ alpha beta)))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.6e+15) {
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (3.0 + (alpha + beta))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.6d+15) then
        tmp = (beta + 1.0d0) / ((beta + 2.0d0) * ((beta + 2.0d0) * (3.0d0 + (alpha + beta))))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.6e+15) {
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (3.0 + (alpha + beta))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.6e+15:
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (3.0 + (alpha + beta))))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.6e+15)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(3.0 + Float64(alpha + beta)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.6e+15)
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (3.0 + (alpha + beta))));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.6e+15], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6e15

   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/ 98.8%

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

      2. +-commutative 98.8%

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

      3. associate-+l+ 98.8%

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

      4. *-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-+l+ 98.8%

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

      7. metadata-eval 98.8%

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

      8. +-commutative 98.8%

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

      9. +-commutative 98.8%

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

      10. +-commutative 98.8%

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

      11. metadata-eval 98.8%

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

      12. metadata-eval 98.8%

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

      13. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in alpha around 0 80.6%

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

      1. +-commutative 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in alpha around 0 65.6%

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

      1. *-un-lft-identity 65.6%

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

      2. associate-+r+ 65.6%

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

      3. +-commutative 65.6%

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

      4. +-commutative 65.6%

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

      5. *-commutative 65.6%

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

   10. Applied egg-rr 65.6%

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

       1. *-lft-identity 65.6%

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

       2. associate-/l/ 65.6%

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

       3. +-commutative 65.6%

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

       4. +-commutative 65.6%

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

       5. *-commutative 65.6%

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

       6. +-commutative 65.6%

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

       7. associate-+r+ 65.6%

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

       8. +-commutative 65.6%

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

       9. +-commutative 65.6%

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

   12. Simplified 65.6%

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

   if 1.6e15 < beta

   1. Initial program 80.5%

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

   3. Taylor expanded in beta around inf 82.0%

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

   4. Taylor expanded in alpha around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. associate-+r+ 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. +-commutative 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6e+16)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (+ 6.0 (* beta (+ beta 5.0))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+16) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d+16) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / (6.0d0 + (beta * (beta + 5.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+16) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6e+16:
		tmp = ((beta + 1.0) / (beta + 2.0)) / (6.0 + (beta * (beta + 5.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6e+16)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(6.0 + Float64(beta * Float64(beta + 5.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6e+16)
		tmp = ((beta + 1.0) / (beta + 2.0)) / (6.0 + (beta * (beta + 5.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.6e+16], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6e16

   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/ 98.8%

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

      2. +-commutative 98.8%

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

      3. associate-+l+ 98.8%

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

      4. *-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-+l+ 98.8%

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

      7. metadata-eval 98.8%

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

      8. +-commutative 98.8%

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

      9. +-commutative 98.8%

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

      10. +-commutative 98.8%

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

      11. metadata-eval 98.8%

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

      12. metadata-eval 98.8%

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

      13. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in alpha around 0 80.6%

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

      1. +-commutative 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in alpha around 0 64.4%

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

   9. Taylor expanded in beta around 0 64.4%

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

       1. +-commutative 64.4%

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

   11. Simplified 64.4%

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

   if 3.6e16 < beta

   1. Initial program 80.5%

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

   3. Taylor expanded in beta around inf 82.0%

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

   4. Taylor expanded in alpha around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. associate-+r+ 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. +-commutative 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4d+16) then
        tmp = (beta + 1.0d0) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4e+16)
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e16

   1. Initial program 99.8%

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

      1. associate-/l/ 98.8%

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

      2. +-commutative 98.8%

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

      3. associate-+l+ 98.8%

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

      4. *-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-+l+ 98.8%

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

      7. metadata-eval 98.8%

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

      8. +-commutative 98.8%

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

      9. +-commutative 98.8%

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

      10. +-commutative 98.8%

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

      11. metadata-eval 98.8%

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

      12. metadata-eval 98.8%

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

      13. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in alpha around 0 80.6%

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

      1. +-commutative 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in alpha around 0 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. associate-/l/ 64.4%

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

      3. +-commutative 64.4%

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

      4. +-commutative 64.4%

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

   10. Applied egg-rr 64.4%

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

       1. *-lft-identity 64.4%

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

       2. +-commutative 64.4%

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

       3. *-commutative 64.4%

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

       4. +-commutative 64.4%

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

   12. Simplified 64.4%

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

   if 4e16 < beta

   1. Initial program 80.5%

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

   3. Taylor expanded in beta around inf 82.0%

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

   4. Taylor expanded in alpha around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. associate-+r+ 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. +-commutative 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 70.1%

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

Alternative 10: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (/ (+ 0.5 (* beta 0.25)) (* (+ beta 2.0) (+ 3.0 (+ alpha beta))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.4d0) then
        tmp = (0.5d0 + (beta * 0.25d0)) / ((beta + 2.0d0) * (3.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.4:
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (3.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.4)
		tmp = Float64(Float64(0.5 + Float64(beta * 0.25)) / Float64(Float64(beta + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4)
		tmp = (0.5 + (beta * 0.25)) / ((beta + 2.0) * (3.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.4], N[(N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 65.0%

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

   9. Taylor expanded in beta around 0 64.6%

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(2 + \beta\right)} \]

   if 4.4000000000000004 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around 0 78.1%

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

      1. +-commutative 78.1%

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

      2. associate-+r+ 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. +-commutative 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.5 + \beta \cdot 0.25}{\left(\beta + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (*
     beta
     (-
      (* beta (- (* beta 0.024691358024691357) 0.011574074074074073))
      0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(beta * 0.024691358024691357), $MachinePrecision] - 0.011574074074074073), $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.4%

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

   if 1.69999999999999996 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around 0 78.1%

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

      1. +-commutative 78.1%

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

      2. associate-+r+ 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. +-commutative 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 7.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 7.6e+158)
     (/ (/ 1.0 beta) (+ beta 3.0))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 7.6e+158) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (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 <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 7.6d+158) then
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    else
        tmp = (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 <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 7.6e+158) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 7.6e+158:
		tmp = (1.0 / beta) / (beta + 3.0)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 7.6e+158)
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 7.6e+158)
		tmp = (1.0 / beta) / (beta + 3.0);
	else
		tmp = (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, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 7.6e+158], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.3%

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

       1. *-commutative 63.3%

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

   11. Simplified 63.3%

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

   if 2.5 < beta < 7.5999999999999997e158

   1. Initial program 86.9%

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

   3. Taylor expanded in beta around inf 63.7%

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

   4. Taylor expanded in alpha around 0 55.9%

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

      1. associate-/r* 57.3%

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

   6. Simplified 57.3%

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

   if 7.5999999999999997e158 < beta

   1. Initial program 76.0%

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

   3. Taylor expanded in beta around inf 94.5%

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

   4. Taylor expanded in alpha around 0 94.5%

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

      1. +-commutative 94.5%

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

      2. associate-+r+ 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. +-commutative 94.5%

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

   6. Simplified 94.5%

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

   7. Taylor expanded in alpha around inf 93.5%

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

   8. Taylor expanded in beta around inf 93.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 7.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (if (<= beta 1e+154)
     (/ 1.0 (* beta (+ beta 3.0)))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 1e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (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 <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else if (beta <= 1d+154) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = (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 <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else if (beta <= 1e+154) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	elif beta <= 1e+154:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	elseif (beta <= 1e+154)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	elseif (beta <= 1e+154)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (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, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1e+154], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.3%

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

       1. *-commutative 63.3%

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

   11. Simplified 63.3%

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

   if 2.5 < beta < 1.00000000000000004e154

   1. Initial program 86.7%

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

   3. Taylor expanded in beta around inf 62.9%

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

   4. Taylor expanded in alpha around 0 56.3%

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

   if 1.00000000000000004e154 < beta

   1. Initial program 76.6%

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

   3. Taylor expanded in beta around inf 94.6%

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

   4. Taylor expanded in alpha around 0 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-+r+ 94.6%

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

      3. +-commutative 94.6%

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

      4. +-commutative 94.6%

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

      5. +-commutative 94.6%

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

   6. Simplified 94.6%

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

   7. Taylor expanded in alpha around inf 92.1%

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

   8. Taylor expanded in beta around inf 92.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.4%

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

   if 1.5 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around 0 78.1%

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

      1. +-commutative 78.1%

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

      2. associate-+r+ 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. +-commutative 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.55d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.55], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.4%

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

   if 1.55000000000000004 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around 0 77.8%

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

      1. +-commutative 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.65)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.65d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.65:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.65)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.65)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.65], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6499999999999999

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.4%

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

   if 1.6499999999999999 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around 0 78.1%

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

      1. +-commutative 78.1%

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

      2. associate-+r+ 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. +-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in beta around inf 77.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.3%

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

       1. *-commutative 63.3%

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

   11. Simplified 63.3%

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

   if 2.7999999999999998 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around 0 78.1%

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

      1. +-commutative 78.1%

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

      2. associate-+r+ 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. +-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in beta around inf 77.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+24}:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.9e+24)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9e+24) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.9d+24) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9e+24) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.9e+24:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.9e+24)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.9e+24)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.9e+24], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.90000000000000008e24

   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/ 98.8%

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

      2. +-commutative 98.8%

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

      3. associate-+l+ 98.8%

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

      4. *-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-+l+ 98.8%

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

      7. metadata-eval 98.8%

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

      8. +-commutative 98.8%

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

      9. +-commutative 98.8%

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

      10. +-commutative 98.8%

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

      11. metadata-eval 98.8%

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

      12. metadata-eval 98.8%

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

      13. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in alpha around 0 80.9%

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

      1. +-commutative 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in alpha around 0 65.7%

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

   9. Taylor expanded in beta around 0 61.1%

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

   10. Taylor expanded in alpha around 0 59.3%

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

       1. *-commutative 59.3%

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

   12. Simplified 59.3%

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

   if 1.90000000000000008e24 < beta

   1. Initial program 79.8%

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

   3. Taylor expanded in beta around inf 82.5%

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

   4. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. associate-+r+ 82.5%

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

      3. +-commutative 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in alpha around inf 54.3%

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

   8. Taylor expanded in beta around inf 54.1%

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

Alternative 19: 47.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   1. Initial program 99.8%

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

      1. associate-/l/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 63.7%

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

   9. Taylor expanded in beta around 0 63.3%

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

       1. *-commutative 63.3%

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

   11. Simplified 63.3%

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

   if 2.89999999999999991 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around inf 7.5%

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

Alternative 20: 47.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10.0)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 10.0d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 10.0:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10.0)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 10.0)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 10.0], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 10

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 65.0%

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

   9. Taylor expanded in beta around 0 63.9%

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

   10. Taylor expanded in alpha around 0 62.0%

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

       1. *-commutative 62.0%

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

   12. Simplified 62.0%

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

   if 10 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around inf 7.5%

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

Alternative 21: 47.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   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/ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. *-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-+l+ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in alpha around 0 65.0%

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

   9. Taylor expanded in beta around 0 63.9%

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

   10. Taylor expanded in alpha around 0 62.6%

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

   if 12 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf 78.1%

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

   4. Taylor expanded in alpha around inf 7.5%

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

Alternative 22: 45.3% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. associate-/l/ 92.0%

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

   2. +-commutative 92.0%

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

   3. associate-+l+ 92.0%

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

   4. *-commutative 92.0%

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

   5. metadata-eval 92.0%

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

   6. associate-+l+ 92.0%

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

   7. metadata-eval 92.0%

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

   8. +-commutative 92.0%

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

   9. +-commutative 92.0%

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

   10. +-commutative 92.0%

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

   11. metadata-eval 92.0%

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

   12. metadata-eval 92.0%

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

   13. associate-+l+ 92.0%

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

3. Simplified 92.0%

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

5. Taylor expanded in alpha around 0 81.1%

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

   1. +-commutative 81.1%

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

7. Simplified 81.1%

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

8. Taylor expanded in alpha around 0 70.2%

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

9. Taylor expanded in beta around 0 43.9%

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

10. Taylor expanded in alpha around 0 42.6%

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

Reproduce

herbie shell --seed 2024146 
(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)))