Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.5%

Time: 13.7s

Alternatives: 24

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(t\_0 + -1\right) \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\left(\alpha + \frac{1}{\beta}\right) + \frac{\alpha}{\beta}\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

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 3.0))))
   (if (<= beta 2.3e+22)
     (/
      (* (+ alpha 1.0) (+ beta 1.0))
      (* (+ t_0 -1.0) (* t_0 (+ alpha (+ beta 2.0)))))
     (/
      (/
       (+
        (+ 1.0 (+ (+ alpha (/ 1.0 beta)) (/ alpha beta)))
        (* (/ (+ 4.0 (* alpha 2.0)) beta) (- -1.0 alpha)))
       beta)
      (+ 1.0 (+ 2.0 (+ beta alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2.3e+22) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 + -1.0) * (t_0 * (alpha + (beta + 2.0))));
	} else {
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (beta <= 2.3d+22) then
        tmp = ((alpha + 1.0d0) * (beta + 1.0d0)) / ((t_0 + (-1.0d0)) * (t_0 * (alpha + (beta + 2.0d0))))
    else
        tmp = (((1.0d0 + ((alpha + (1.0d0 / beta)) + (alpha / beta))) + (((4.0d0 + (alpha * 2.0d0)) / beta) * ((-1.0d0) - alpha))) / beta) / (1.0d0 + (2.0d0 + (beta + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2.3e+22) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 + -1.0) * (t_0 * (alpha + (beta + 2.0))));
	} else {
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 2.3e+22:
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 + -1.0) * (t_0 * (alpha + (beta + 2.0))))
	else:
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (beta + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 2.3e+22)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / Float64(Float64(t_0 + -1.0) * Float64(t_0 * Float64(alpha + Float64(beta + 2.0)))));
	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(beta + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 2.3e+22)
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 + -1.0) * (t_0 * (alpha + (beta + 2.0))));
	else
		tmp = (((1.0 + ((alpha + (1.0 / beta)) + (alpha / beta))) + (((4.0 + (alpha * 2.0)) / beta) * (-1.0 - alpha))) / beta) / (1.0 + (2.0 + (beta + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.3e+22], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + -1.0), $MachinePrecision] * N[(t$95$0 * N[(alpha + N[(beta + 2.0), $MachinePrecision]), $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[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3000000000000002e22

   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 96.0%

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

      1. expm1-log1p-u 93.8%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      2. log1p-define 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      3. +-commutative 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      4. associate-+r+ 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      5. associate-+l+ 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      6. metadata-eval 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      7. expm1-undefine 94.3%

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

      8. add-exp-log 96.0%

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

      9. associate-+r+ 96.0%

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

   6. Applied egg-rr 96.0%

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

   if 2.3000000000000002e22 < beta

   1. Initial program 79.0%

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

   3. Taylor expanded in beta around inf 74.9%

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

      1. associate-+r+ 74.9%

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

      2. associate-/l* 79.3%

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

   5. Simplified 79.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \left(\beta + 3\right)\right) + -1\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\left(\alpha + \frac{1}{\beta}\right) + \frac{\alpha}{\beta}\right)\right) + \frac{4 + \alpha \cdot 2}{\beta} \cdot \left(-1 - \alpha\right)}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+131}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{t\_0}}{t\_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+131)
     (/
      (/ (+ 1.0 (+ alpha (+ beta (* beta alpha)))) t_0)
      (* t_0 (+ 3.0 (+ beta alpha))))
     (* (/ (+ alpha 1.0) t_0) (/ (- 1.0 (* 2.0 (/ alpha beta))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+131) {
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1d+131) then
        tmp = ((1.0d0 + (alpha + (beta + (beta * alpha)))) / t_0) / (t_0 * (3.0d0 + (beta + alpha)))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+131) {
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1e+131:
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / (t_0 * (3.0 + (beta + alpha)))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1e+131)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + Float64(beta + Float64(beta * alpha)))) / t_0) / Float64(t_0 * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1e+131)
		tmp = ((1.0 + (alpha + (beta + (beta * alpha)))) / t_0) / (t_0 * (3.0 + (beta + alpha)));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+131], N[(N[(N[(1.0 + N[(alpha + N[(beta + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999991e130

   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/ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+l+ 97.7%

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

      4. *-commutative 97.7%

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

      5. metadata-eval 97.7%

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

      6. associate-+l+ 97.7%

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

      7. metadata-eval 97.7%

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

      8. +-commutative 97.7%

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

      9. +-commutative 97.7%

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

      10. +-commutative 97.7%

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

      11. metadata-eval 97.7%

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

      12. metadata-eval 97.7%

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

      13. associate-+l+ 97.7%

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

   3. Simplified 97.7%

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

   if 9.9999999999999991e130 < beta

   1. Initial program 68.6%

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

   3. Simplified 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 76.9%

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

      2. +-commutative 76.9%

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

   6. Applied egg-rr 76.9%

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

   7. Taylor expanded in beta around inf 80.3%

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

      1. mul-1-neg 80.3%

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

   9. Simplified 80.3%

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

   10. Taylor expanded in alpha around inf 80.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+131}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(t\_1 + -1\right) \cdot \left(t\_1 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (+ alpha (+ beta 3.0))))
   (if (<= beta 3.2e+22)
     (/ (* (+ alpha 1.0) (+ beta 1.0)) (* (+ t_1 -1.0) (* t_1 t_0)))
     (* (/ (+ alpha 1.0) t_0) (/ (- 1.0 (* 2.0 (/ alpha beta))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.2e+22) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_1 + -1.0) * (t_1 * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = alpha + (beta + 3.0d0)
    if (beta <= 3.2d+22) then
        tmp = ((alpha + 1.0d0) * (beta + 1.0d0)) / ((t_1 + (-1.0d0)) * (t_1 * t_0))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.2e+22) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_1 + -1.0) * (t_1 * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 3.2e+22:
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_1 + -1.0) * (t_1 * t_0))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 3.2e+22)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / Float64(Float64(t_1 + -1.0) * Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 3.2e+22)
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_1 + -1.0) * (t_1 * t_0));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.2e+22], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 + -1.0), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e22

   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 96.0%

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

      1. expm1-log1p-u 93.8%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      2. log1p-define 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      3. +-commutative 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      4. associate-+r+ 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      5. associate-+l+ 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      6. metadata-eval 95.2%

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \]

      7. expm1-undefine 94.3%

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

      8. add-exp-log 96.0%

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

      9. associate-+r+ 96.0%

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

   6. Applied egg-rr 96.0%

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

   if 3.2e22 < beta

   1. Initial program 79.0%

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

   3. Simplified 59.2%

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

      1. times-frac 82.9%

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

      2. +-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in beta around inf 79.4%

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

      1. mul-1-neg 79.4%

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

   9. Simplified 79.4%

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

   10. Taylor expanded in alpha around inf 79.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \left(\beta + 3\right)\right) + -1\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{t\_0 \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+20)
     (/ (* (+ alpha 1.0) (+ beta 1.0)) (* t_0 (* (+ alpha (+ beta 3.0)) t_0)))
     (* (/ (+ alpha 1.0) t_0) (/ (- 1.0 (* 2.0 (/ alpha beta))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+20) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2d+20) then
        tmp = ((alpha + 1.0d0) * (beta + 1.0d0)) / (t_0 * ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+20) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2e+20:
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+20)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / Float64(t_0 * Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2e+20)
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+20], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e20

   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 96.0%

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

   if 2e20 < beta

   1. Initial program 79.0%

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

   3. Simplified 59.2%

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

      1. times-frac 82.9%

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

      2. +-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in beta around inf 79.4%

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

      1. mul-1-neg 79.4%

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

   9. Simplified 79.4%

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

   10. Taylor expanded in alpha around inf 79.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha + 1}{t\_0}\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+141}:\\ \;\;\;\;t\_1 \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ (+ alpha 1.0) t_0)))
   (if (<= beta 5e+141)
     (* t_1 (/ (+ beta 1.0) (* (+ alpha (+ beta 3.0)) t_0)))
     (* t_1 (/ (- 1.0 (* 2.0 (/ alpha 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 <= 5e+141) {
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = t_1 * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (alpha + 1.0d0) / t_0
    if (beta <= 5d+141) then
        tmp = t_1 * ((beta + 1.0d0) / ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = t_1 * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + 1.0) / t_0;
	double tmp;
	if (beta <= 5e+141) {
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = t_1 * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (alpha + 1.0) / t_0
	tmp = 0
	if beta <= 5e+141:
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = t_1 * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(alpha + 1.0) / t_0)
	tmp = 0.0
	if (beta <= 5e+141)
		tmp = Float64(t_1 * Float64(Float64(beta + 1.0) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (alpha + 1.0) / t_0;
	tmp = 0.0;
	if (beta <= 5e+141)
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = t_1 * ((1.0 - (2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 5e+141], N[(t$95$1 * N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000025e141

   1. Initial program 99.3%

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

   3. Simplified 90.8%

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

      1. times-frac 98.2%

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

      2. +-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   if 5.00000000000000025e141 < beta

   1. Initial program 66.4%

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

   3. Simplified 59.4%

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

      1. times-frac 75.2%

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

      2. +-commutative 75.2%

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

   6. Applied egg-rr 75.2%

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

   7. Taylor expanded in beta around inf 78.9%

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

      1. mul-1-neg 78.9%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in alpha around inf 78.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 195000000:\\ \;\;\;\;\frac{\beta + 1}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 195000000.0)
   (/ (+ beta 1.0) (+ 12.0 (* beta (+ 16.0 (* beta (+ beta 7.0))))))
   (*
    (/ (+ 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 <= 195000000.0) {
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} 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 <= 195000000.0d0) then
        tmp = (beta + 1.0d0) / (12.0d0 + (beta * (16.0d0 + (beta * (beta + 7.0d0)))))
    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 <= 195000000.0) {
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} 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 <= 195000000.0:
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))))
	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 <= 195000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(beta + 7.0))))));
	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 <= 195000000.0)
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	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, 195000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(beta + 7.0), $MachinePrecision]), $MachinePrecision]), $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 < 1.95e8

   1. Initial program 99.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in beta around -inf 94.4%

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

      1. associate-*r* 94.4%

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

      2. mul-1-neg 94.4%

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

      3. sub-neg 94.4%

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

      4. associate-*r/ 94.4%

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

      5. distribute-lft-in 94.4%

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

      6. metadata-eval 94.4%

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

      7. mul-1-neg 94.4%

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

      8. unsub-neg 94.4%

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

      9. metadata-eval 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in alpha around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. +-commutative 67.2%

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

      3. +-commutative 67.2%

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

      4. associate-*r/ 67.2%

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

      5. metadata-eval 67.2%

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

   10. Simplified 67.2%

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

   11. Taylor expanded in beta around 0 67.6%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 67.6%

          \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \color{blue}{\left(\beta + 7\right)}\right)} \]

   13. Simplified 67.6%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}} \]

   if 1.95e8 < beta

   1. Initial program 81.3%

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

   3. Simplified 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 84.7%

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

      2. +-commutative 84.7%

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

   6. Applied egg-rr 84.7%

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

   7. Taylor expanded in beta around inf 81.6%

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

      1. mul-1-neg 81.6%

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

   9. Simplified 81.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 195000000:\\ \;\;\;\;\frac{\beta + 1}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4 + \alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{t\_0 \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3e+22)
     (/ (* (+ alpha 1.0) (+ beta 1.0)) (* t_0 (* (+ beta 3.0) (+ beta 2.0))))
     (* (/ (+ alpha 1.0) t_0) (/ (- 1.0 (* 2.0 (/ alpha beta))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3e+22) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3d+22) then
        tmp = ((alpha + 1.0d0) * (beta + 1.0d0)) / (t_0 * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3e+22) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3e+22:
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3e+22)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / Float64(t_0 * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3e+22)
		tmp = ((alpha + 1.0) * (beta + 1.0)) / (t_0 * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3e+22], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e22

   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 96.0%

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

   5. Taylor expanded in alpha around 0 69.1%

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

      1. +-commutative 69.1%

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

      2. +-commutative 69.1%

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

   7. Simplified 69.1%

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

   if 3e22 < beta

   1. Initial program 79.0%

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

   3. Simplified 59.2%

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

      1. times-frac 82.9%

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

      2. +-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in beta around inf 79.4%

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

      1. mul-1-neg 79.4%

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

   9. Simplified 79.4%

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

   10. Taylor expanded in alpha around inf 79.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+141}:\\ \;\;\;\;t\_0 \cdot \frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))))
   (if (<= beta 5e+141)
     (* t_0 (/ (+ beta 1.0) (* (+ beta 3.0) (+ beta 2.0))))
     (* t_0 (/ (- 1.0 (* 2.0 (/ alpha beta))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 5e+141) {
		tmp = t_0 * ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = t_0 * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 1.0d0) / (alpha + (beta + 2.0d0))
    if (beta <= 5d+141) then
        tmp = t_0 * ((beta + 1.0d0) / ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = t_0 * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 5e+141) {
		tmp = t_0 * ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = t_0 * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0))
	tmp = 0
	if beta <= 5e+141:
		tmp = t_0 * ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = t_0 * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0)))
	tmp = 0.0
	if (beta <= 5e+141)
		tmp = Float64(t_0 * Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	tmp = 0.0;
	if (beta <= 5e+141)
		tmp = t_0 * ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = t_0 * ((1.0 - (2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+141], N[(t$95$0 * N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000025e141

   1. Initial program 99.3%

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

   3. Simplified 90.8%

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

      1. times-frac 98.2%

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

      2. +-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in alpha around 0 70.6%

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

      1. +-commutative 70.6%

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

      2. +-commutative 70.6%

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

   9. Simplified 70.6%

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

   if 5.00000000000000025e141 < beta

   1. Initial program 66.4%

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

   3. Simplified 59.4%

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

      1. times-frac 75.2%

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

      2. +-commutative 75.2%

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

   6. Applied egg-rr 75.2%

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

   7. Taylor expanded in beta around inf 78.9%

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

      1. mul-1-neg 78.9%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in alpha around inf 78.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+141)
   (*
    (/ (+ beta 1.0) (* (+ beta 3.0) (+ beta 2.0)))
    (/ (+ alpha 1.0) (+ beta 2.0)))
   (*
    (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))
    (/ (- 1.0 (* 2.0 (/ alpha beta))) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+141) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) * ((alpha + 1.0) / (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+141) then
        tmp = ((beta + 1.0d0) / ((beta + 3.0d0) * (beta + 2.0d0))) * ((alpha + 1.0d0) / (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * ((1.0d0 - (2.0d0 * (alpha / beta))) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+141) {
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) * ((alpha + 1.0) / (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5e+141:
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) * ((alpha + 1.0) / (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * (alpha / beta))) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+141)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0))) * Float64(Float64(alpha + 1.0) / Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(2.0 * Float64(alpha / beta))) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+141)
		tmp = ((beta + 1.0) / ((beta + 3.0) * (beta + 2.0))) * ((alpha + 1.0) / (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (2.0 * (alpha / beta))) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5e+141], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000025e141

   1. Initial program 99.3%

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

   3. Simplified 90.8%

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

      1. times-frac 98.2%

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

      2. +-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in alpha around 0 70.6%

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

      1. +-commutative 70.6%

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

      2. +-commutative 70.6%

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

   9. Simplified 70.6%

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

   10. Taylor expanded in alpha around 0 70.3%

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

       1. +-commutative 70.3%

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

   12. Simplified 70.3%

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

   if 5.00000000000000025e141 < beta

   1. Initial program 66.4%

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

   3. Simplified 59.4%

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

      1. times-frac 75.2%

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

      2. +-commutative 75.2%

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

   6. Applied egg-rr 75.2%

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

   7. Taylor expanded in beta around inf 78.9%

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

      1. mul-1-neg 78.9%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in alpha around inf 78.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - 2 \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 250000000:\\ \;\;\;\;\frac{\beta + 1}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 250000000.0)
   (/ (+ beta 1.0) (+ 12.0 (* beta (+ 16.0 (* beta (+ beta 7.0))))))
   (* (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))) (/ (- 1.0 (/ 4.0 beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 250000000.0) {
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 250000000.0d0) then
        tmp = (beta + 1.0d0) / (12.0d0 + (beta * (16.0d0 + (beta * (beta + 7.0d0)))))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * ((1.0d0 - (4.0d0 / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 250000000.0) {
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 250000000.0:
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 250000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(beta + 7.0))))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(4.0 / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 250000000.0)
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 250000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(beta + 7.0), $MachinePrecision]), $MachinePrecision]), $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[(4.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5e8

   1. Initial program 99.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in beta around -inf 94.4%

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

      1. associate-*r* 94.4%

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

      2. mul-1-neg 94.4%

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

      3. sub-neg 94.4%

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

      4. associate-*r/ 94.4%

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

      5. distribute-lft-in 94.4%

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

      6. metadata-eval 94.4%

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

      7. mul-1-neg 94.4%

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

      8. unsub-neg 94.4%

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

      9. metadata-eval 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in alpha around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. +-commutative 67.2%

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

      3. +-commutative 67.2%

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

      4. associate-*r/ 67.2%

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

      5. metadata-eval 67.2%

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

   10. Simplified 67.2%

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

   11. Taylor expanded in beta around 0 67.6%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 67.6%

          \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \color{blue}{\left(\beta + 7\right)}\right)} \]

   13. Simplified 67.6%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}} \]

   if 2.5e8 < beta

   1. Initial program 81.3%

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

   3. Simplified 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 84.7%

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

      2. +-commutative 84.7%

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

   6. Applied egg-rr 84.7%

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

   7. Taylor expanded in alpha around 0 78.7%

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

      1. +-commutative 78.7%

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

      2. +-commutative 78.7%

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

   9. Simplified 78.7%

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

   10. Taylor expanded in beta around inf 82.0%

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

       1. associate-*r/ 82.0%

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

       2. metadata-eval 82.0%

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

   12. Simplified 82.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 250000000:\\ \;\;\;\;\frac{\beta + 1}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 90000000:\\ \;\;\;\;\frac{\beta + 1}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + 2} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 90000000.0)
   (/ (+ beta 1.0) (+ 12.0 (* beta (+ 16.0 (* beta (+ beta 7.0))))))
   (* (/ (+ alpha 1.0) (+ beta 2.0)) (/ (- 1.0 (/ 4.0 beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 90000000.0) {
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 90000000.0d0) then
        tmp = (beta + 1.0d0) / (12.0d0 + (beta * (16.0d0 + (beta * (beta + 7.0d0)))))
    else
        tmp = ((alpha + 1.0d0) / (beta + 2.0d0)) * ((1.0d0 - (4.0d0 / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 90000000.0) {
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	} else {
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 90000000.0:
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))))
	else:
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 90000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(beta + 7.0))))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + 2.0)) * Float64(Float64(1.0 - Float64(4.0 / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 90000000.0)
		tmp = (beta + 1.0) / (12.0 + (beta * (16.0 + (beta * (beta + 7.0)))));
	else
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 90000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(beta + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(4.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9e7

   1. Initial program 99.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in beta around -inf 94.4%

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

      1. associate-*r* 94.4%

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

      2. mul-1-neg 94.4%

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

      3. sub-neg 94.4%

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

      4. associate-*r/ 94.4%

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

      5. distribute-lft-in 94.4%

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

      6. metadata-eval 94.4%

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

      7. mul-1-neg 94.4%

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

      8. unsub-neg 94.4%

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

      9. metadata-eval 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in alpha around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. +-commutative 67.2%

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

      3. +-commutative 67.2%

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

      4. associate-*r/ 67.2%

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

      5. metadata-eval 67.2%

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

   10. Simplified 67.2%

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

   11. Taylor expanded in beta around 0 67.6%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 67.6%

          \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \color{blue}{\left(\beta + 7\right)}\right)} \]

   13. Simplified 67.6%

       \[\leadsto \frac{1 + \beta}{\color{blue}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}} \]

   if 9e7 < beta

   1. Initial program 81.3%

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

   3. Simplified 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 84.7%

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

      2. +-commutative 84.7%

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

   6. Applied egg-rr 84.7%

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

   7. Taylor expanded in alpha around 0 78.7%

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

      1. +-commutative 78.7%

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

      2. +-commutative 78.7%

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

   9. Simplified 78.7%

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

   10. Taylor expanded in alpha around 0 78.6%

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

       1. +-commutative 78.6%

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

   12. Simplified 78.6%

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

   13. Taylor expanded in beta around inf 81.8%

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

       1. associate-*r/ 82.0%

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

       2. metadata-eval 82.0%

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

   15. Simplified 81.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 90000000:\\ \;\;\;\;\frac{\beta + 1}{12 + \beta \cdot \left(16 + \beta \cdot \left(\beta + 7\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + 2} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + 2} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0)
   (+
    0.08333333333333333
    (*
     beta
     (-
      (* beta (- (* beta 0.024691358024691357) 0.011574074074074073))
      0.027777777777777776)))
   (* (/ (+ alpha 1.0) (+ beta 2.0)) (/ (- 1.0 (/ 4.0 beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / (beta + 2.0d0)) * ((1.0d0 - (4.0d0 / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta)
	return tmp

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / (beta + 2.0)) * ((1.0 - (4.0 / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 68.1%

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

   if 4 < beta

   1. Initial program 82.8%

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

   3. Simplified 66.6%

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

      1. times-frac 85.9%

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

      2. +-commutative 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in alpha around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. +-commutative 76.0%

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

   9. Simplified 76.0%

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

   10. Taylor expanded in alpha around 0 75.9%

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

       1. +-commutative 75.9%

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

   12. Simplified 75.9%

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

   13. Taylor expanded in beta around inf 77.9%

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

       1. associate-*r/ 78.2%

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

       2. metadata-eval 78.2%

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

   15. Simplified 77.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta + 2} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.4% accurate, 1.9× speedup

Math

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

FPCore

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

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) * (1.0 / (alpha + (beta + 3.0)))) / 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 <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / (alpha + (beta + 3.0d0)))) / beta
    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) * (1.0 / (alpha + (beta + 3.0)))) / beta;
	}
	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) * (1.0 / (alpha + (beta + 3.0)))) / beta
	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) * Float64(1.0 / Float64(alpha + Float64(beta + 3.0)))) / beta);
	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) * (1.0 / (alpha + (beta + 3.0)))) / 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, 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] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 68.1%

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

   if 1.69999999999999996 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

      1. div-inv 75.8%

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

      2. +-commutative 75.8%

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

      3. metadata-eval 75.8%

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

      4. associate-+l+ 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-+r+ 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. associate-*l/ 75.9%

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

   7. Applied egg-rr 75.9%

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

4. Final simplification 70.7%

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

Alternative 14: 97.4% 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%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 68.1%

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

   if 1.69999999999999996 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

      1. div-inv 75.8%

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

      2. +-commutative 75.8%

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

      3. metadata-eval 75.8%

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

      4. associate-+l+ 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-+r+ 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. *-lft-identity 75.9%

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

      4. +-commutative 75.9%

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

      5. +-commutative 75.9%

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

      6. +-commutative 75.9%

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

      7. +-commutative 75.9%

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

   7. Simplified 75.9%

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

4. Final simplification 70.7%

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

Alternative 15: 97.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.52)
   (+
    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 <= 1.52) {
		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 <= 1.52d0) 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 <= 1.52) {
		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 <= 1.52:
		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 <= 1.52)
		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 <= 1.52)
		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, 1.52], 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 < 1.52

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.9%

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

   if 1.52 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

      1. div-inv 75.8%

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

      2. +-commutative 75.8%

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

      3. metadata-eval 75.8%

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

      4. associate-+l+ 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-+r+ 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. *-lft-identity 75.9%

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

      4. +-commutative 75.9%

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

      5. +-commutative 75.9%

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

      6. +-commutative 75.9%

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

      7. +-commutative 75.9%

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

   7. Simplified 75.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.3% 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%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.9%

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

   if 1.55000000000000004 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

   4. Taylor expanded in alpha around 0 75.7%

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

      1. +-commutative 75.7%

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

   6. Simplified 75.7%

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

4. Final simplification 70.6%

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

Alternative 17: 92.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / (beta + 2.0)) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = (1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / (beta + 2.0)) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.2:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = (1.0 / (beta + 2.0)) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = (1.0 / (beta + 2.0)) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.2], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.9%

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

   if 1.19999999999999996 < beta

   1. Initial program 82.8%

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

      1. associate-/l/ 77.4%

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

      2. +-commutative 77.4%

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

      3. associate-+l+ 77.4%

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

      4. *-commutative 77.4%

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

      5. metadata-eval 77.4%

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

      6. associate-+l+ 77.4%

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

      7. metadata-eval 77.4%

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

      8. +-commutative 77.4%

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

      9. +-commutative 77.4%

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

      10. +-commutative 77.4%

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

      11. metadata-eval 77.4%

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

      12. metadata-eval 77.4%

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

      13. associate-+l+ 77.4%

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

   3. Simplified 77.4%

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

      1. clear-num 77.4%

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

      2. inv-pow 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-rgt1-in 77.4%

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

      5. fma-define 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. unpow-1 77.4%

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

      2. +-commutative 77.4%

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

      3. +-commutative 77.4%

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

      4. +-commutative 77.4%

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

      5. +-commutative 77.4%

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

      6. fma-undefine 77.4%

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

      7. +-commutative 77.4%

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

      8. *-commutative 77.4%

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

      9. +-commutative 77.4%

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

      10. associate-+r+ 77.4%

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

      11. distribute-lft1-in 77.4%

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

      12. +-commutative 77.4%

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

      13. +-commutative 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in beta around inf 78.8%

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

   10. Taylor expanded in alpha around 0 69.4%

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

       1. associate-/r* 70.2%

          \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

       2. +-commutative 70.2%

          \[\leadsto \frac{\frac{1}{\color{blue}{\beta + 2}}}{3 + \beta} \]

       3. +-commutative 70.2%

          \[\leadsto \frac{\frac{1}{\beta + 2}}{\color{blue}{\beta + 3}} \]

   12. Simplified 70.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 92.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.52)
   (+
    0.08333333333333333
    (* beta (- (* beta -0.011574074074074073) 0.027777777777777776)))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.52d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * (-0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.52) {
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.52:
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776))
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.52)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * -0.011574074074074073) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.52)
		tmp = 0.08333333333333333 + (beta * ((beta * -0.011574074074074073) - 0.027777777777777776));
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.52], N[(0.08333333333333333 + N[(beta * N[(N[(beta * -0.011574074074074073), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.52

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.9%

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

   if 1.52 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

   4. Taylor expanded in alpha around 0 69.1%

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

      1. associate-/r* 70.0%

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

      2. +-commutative 70.0%

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

   6. Simplified 70.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.52:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 92.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.6%

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

       1. *-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 2.5 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

   4. Taylor expanded in alpha around 0 69.1%

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

      1. associate-/r* 70.0%

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

      2. +-commutative 70.0%

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

   6. Simplified 70.0%

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

Alternative 20: 92.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.6%

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

       1. *-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 2.60000000000000009 < beta

   1. Initial program 82.8%

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

   3. Simplified 66.6%

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

      1. times-frac 85.9%

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

      2. +-commutative 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in beta around inf 77.6%

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

      1. mul-1-neg 77.6%

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

   9. Simplified 77.6%

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

   10. Taylor expanded in alpha around inf 75.2%

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

   11. Taylor expanded in alpha around 0 69.1%

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

       1. associate-/r* 69.9%

          \[\leadsto \color{blue}{\frac{\frac{1}{\beta}}{2 + \beta}} \]

       2. +-commutative 69.9%

          \[\leadsto \frac{\frac{1}{\beta}}{\color{blue}{\beta + 2}} \]

   13. Simplified 69.9%

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

Alternative 21: 91.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.6%

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

       1. *-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 2.5 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

   4. Taylor expanded in alpha around 0 69.1%

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

4. Final simplification 68.1%

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

Alternative 22: 46.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.95)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		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 <= 2.95d0) 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 <= 2.95) {
		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 <= 2.95:
		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 <= 2.95)
		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 <= 2.95)
		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, 2.95], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9500000000000002

   1. Initial program 99.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in beta around -inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. mul-1-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. associate-*r/ 94.2%

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

      5. distribute-lft-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. mul-1-neg 94.2%

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

      8. unsub-neg 94.2%

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

      9. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in alpha around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. associate-*r/ 68.1%

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

      5. metadata-eval 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in beta around 0 67.6%

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

       1. *-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 2.9500000000000002 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 75.9%

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

   4. Taylor expanded in alpha around inf 6.9%

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

Alternative 23: 45.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 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
    code = 0.16666666666666666d0 / (beta + 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

3. Simplified 85.8%

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

   1. times-frac 94.5%

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

   2. +-commutative 94.5%

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

6. Applied egg-rr 94.5%

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

7. Taylor expanded in alpha around 0 71.0%

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

   1. +-commutative 71.0%

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

   2. +-commutative 71.0%

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

9. Simplified 71.0%

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

10. Taylor expanded in beta around 0 46.1%

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

11. Taylor expanded in alpha around 0 46.4%

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

    1. +-commutative 46.4%

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

13. Simplified 46.4%

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

Alternative 24: 43.9% 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 94.0%

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

3. Simplified 85.8%

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

5. Taylor expanded in beta around -inf 84.8%

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

   1. associate-*r* 84.8%

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

   2. mul-1-neg 84.8%

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

   3. sub-neg 84.8%

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

   4. associate-*r/ 84.8%

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

   5. distribute-lft-in 84.8%

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

   6. metadata-eval 84.8%

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

   7. mul-1-neg 84.8%

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

   8. unsub-neg 84.8%

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

   9. metadata-eval 84.8%

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

7. Simplified 84.8%

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

8. Taylor expanded in alpha around 0 67.3%

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

   1. *-commutative 67.3%

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

   2. +-commutative 67.3%

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

   3. +-commutative 67.3%

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

   4. associate-*r/ 67.3%

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

   5. metadata-eval 67.3%

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

10. Simplified 67.3%

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

11. Taylor expanded in beta around 0 45.6%

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

Reproduce

herbie shell --seed 2024170 
(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)))