Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.6%

Time: 27.9s

Alternatives: 15

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{1 + \beta}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 1.35e+15)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_0 t_1)))
     (/
      (*
       (/ (+ 1.0 alpha) (+ (+ alpha 2.0) beta))
       (+ 1.0 (/ (- -1.0 alpha) beta)))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.35e+15) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 1.35d+15) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_0 * t_1))
    else
        tmp = (((1.0d0 + alpha) / ((alpha + 2.0d0) + beta)) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.35e+15) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 1.35e+15:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1))
	else:
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 1.35e+15)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) + beta)) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 1.35e+15)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	else
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.35e+15], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.35e15

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   if 1.35e15 < beta

   1. Initial program 80.2%

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

   3. Simplified 85.7%

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

      1. associate-*l/ 85.7%

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

      2. +-commutative 85.7%

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

      3. associate-+r+ 85.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. associate-+r+ 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 89.0%

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

       1. mul-1-neg 89.0%

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

       2. unsub-neg 89.0%

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

   11. Simplified 89.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{2 + \beta} \cdot \frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.15e+14)
   (* (/ 1.0 (+ 2.0 beta)) (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 2.0 beta))))
   (/
    (*
     (/ (+ 1.0 alpha) (+ (+ alpha 2.0) beta))
     (+ 1.0 (/ (- -1.0 alpha) beta)))
    (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+14) {
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.15d+14) then
        tmp = (1.0d0 / (2.0d0 + beta)) * ((1.0d0 + beta) / ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = (((1.0d0 + alpha) / ((alpha + 2.0d0) + beta)) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+14) {
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.15e+14:
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.15e+14)
		tmp = Float64(Float64(1.0 / Float64(2.0 + beta)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) + beta)) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.15e+14)
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = (((1.0 + alpha) / ((alpha + 2.0) + beta)) * (1.0 + ((-1.0 - alpha) / beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.15e+14], N[(N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15e14

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 63.4%

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

   6. Taylor expanded in alpha around 0 63.5%

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

   if 1.15e14 < beta

   1. Initial program 80.2%

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

   3. Simplified 85.7%

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

      1. associate-*l/ 85.7%

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

      2. +-commutative 85.7%

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

      3. associate-+r+ 85.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. associate-+r+ 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 89.0%

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

       1. mul-1-neg 89.0%

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

       2. unsub-neg 89.0%

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

   11. Simplified 89.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{2 + \beta} \cdot \frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Simplified 95.4%

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

   1. associate-*l/ 95.4%

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

   2. +-commutative 95.4%

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

   3. associate-+r+ 95.4%

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

   4. associate-/r* 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+r+ 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. associate-*r/ 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{2 + \beta} \cdot \frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.05e+16)
   (* (/ 1.0 (+ 2.0 beta)) (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 2.0 beta))))
   (/ (/ (+ 1.0 alpha) (+ beta (+ alpha 3.0))) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.05e+16) {
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.05d+16) then
        tmp = (1.0d0 / (2.0d0 + beta)) * ((1.0d0 + beta) / ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 3.0d0))) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.05e+16) {
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.05e+16:
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.05e+16)
		tmp = Float64(Float64(1.0 / Float64(2.0 + beta)) * Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 3.0))) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.05e+16)
		tmp = (1.0 / (2.0 + beta)) * ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.05e+16], N[(N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.05e16

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 63.6%

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

   6. Taylor expanded in alpha around 0 63.7%

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

   if 1.05e16 < beta

   1. Initial program 79.9%

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

      1. associate-/l/ 74.3%

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

      2. +-commutative 74.3%

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

      3. associate-+l+ 74.3%

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

      4. *-commutative 74.3%

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

      5. metadata-eval 74.3%

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

      6. associate-+l+ 74.3%

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

      7. metadata-eval 74.3%

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

      8. +-commutative 74.3%

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

      9. metadata-eval 74.3%

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

      10. metadata-eval 74.3%

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

      11. associate-+l+ 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in beta around -inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. sub-neg 84.4%

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

      3. mul-1-neg 84.4%

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

      4. distribute-neg-in 84.4%

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

      5. +-commutative 84.4%

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

      6. mul-1-neg 84.4%

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

      7. distribute-lft-in 84.4%

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

      8. metadata-eval 84.4%

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

      9. mul-1-neg 84.4%

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

      10. unsub-neg 84.4%

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

   7. Simplified 84.4%

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

      1. distribute-frac-neg 84.4%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 36.6%

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

      4. sqr-neg 36.6%

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

      5. sqrt-unprod 43.0%

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

      6. add-sqr-sqrt 43.0%

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

      7. associate-/r* 38.1%

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

      8. add-sqr-sqrt 38.1%

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

      9. sqrt-unprod 36.7%

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

      10. sqr-neg 36.7%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 89.3%

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

      13. +-commutative 89.3%

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

      14. +-commutative 89.3%

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

   9. Applied egg-rr 89.3%

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

       1. distribute-neg-frac 89.3%

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

       2. sub-neg 89.3%

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

       3. metadata-eval 89.3%

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

       4. distribute-neg-in 89.3%

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

       5. distribute-neg-frac 89.3%

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

       6. distribute-neg-in 89.3%

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

       7. metadata-eval 89.3%

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

       8. sub-neg 89.3%

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

       9. associate-+r+ 89.3%

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

       10. +-commutative 89.3%

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

       11. associate-+l+ 89.3%

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

       12. +-commutative 89.3%

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

       13. +-commutative 89.3%

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

       14. associate-+l+ 89.3%

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

   11. Simplified 89.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{2 + \beta} \cdot \frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.25:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \beta \cdot 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.25)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ 0.16666666666666666 (* beta 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.25) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (beta * 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.25d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * (0.16666666666666666d0 + (beta * 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.25) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (beta * 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.25:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (beta * 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.25)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(0.16666666666666666 + Float64(beta * 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.25)
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (beta * 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.25], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(beta * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.25

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 63.2%

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

   6. Taylor expanded in beta around 0 63.2%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + 0.027777777777777776 \cdot \beta\right)} \]

   if 3.25 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

      1. associate-*l/ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+r+ 86.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. associate-+r+ 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 85.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.25:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \beta \cdot 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.0d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   9. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   10. Simplified 62.8%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   if 5 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

      1. associate-*l/ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+r+ 86.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. associate-+r+ 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 85.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (+ 1.0 alpha) (+ beta (+ alpha 3.0))) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.7d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((1.0d0 + alpha) / (beta + (alpha + 3.0d0))) / (2.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(alpha + 3.0))) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7)
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((1.0 + alpha) / (beta + (alpha + 3.0))) / (2.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.7], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   9. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   10. Simplified 62.8%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   if 2.7000000000000002 < beta

   1. Initial program 81.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. associate-+l+ 76.4%

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

      4. *-commutative 76.4%

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

      5. metadata-eval 76.4%

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

      6. associate-+l+ 76.4%

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

      7. metadata-eval 76.4%

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

      8. +-commutative 76.4%

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

      9. metadata-eval 76.4%

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

      10. metadata-eval 76.4%

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

      11. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in beta around -inf 83.2%

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

      1. mul-1-neg 83.2%

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

      2. sub-neg 83.2%

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

      3. mul-1-neg 83.2%

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

      4. distribute-neg-in 83.2%

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

      5. +-commutative 83.2%

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

      6. mul-1-neg 83.2%

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

      7. distribute-lft-in 83.2%

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

      8. metadata-eval 83.2%

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

      9. mul-1-neg 83.2%

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

      10. unsub-neg 83.2%

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

   7. Simplified 83.2%

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

      1. distribute-frac-neg 83.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 33.8%

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

      4. sqr-neg 33.8%

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

      5. sqrt-unprod 40.8%

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

      6. add-sqr-sqrt 40.8%

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

      7. associate-/r* 35.3%

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

      8. add-sqr-sqrt 35.3%

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

      9. sqrt-unprod 33.9%

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

      10. sqr-neg 33.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 86.6%

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

      13. +-commutative 86.6%

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

      14. +-commutative 86.6%

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

   9. Applied egg-rr 86.6%

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

       1. distribute-neg-frac 86.6%

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

       2. sub-neg 86.6%

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

       3. metadata-eval 86.6%

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

       4. distribute-neg-in 86.6%

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

       5. distribute-neg-frac 86.6%

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

       6. distribute-neg-in 86.6%

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

       7. metadata-eval 86.6%

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

       8. sub-neg 86.6%

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

       9. associate-+r+ 86.6%

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

       10. +-commutative 86.6%

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

       11. associate-+l+ 86.6%

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

       12. +-commutative 86.6%

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

       13. +-commutative 86.6%

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

       14. associate-+l+ 86.6%

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

   11. Simplified 86.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.3)
   (/ 1.0 (/ (+ 2.0 beta) 0.16666666666666666))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.3d0) then
        tmp = 1.0d0 / ((2.0d0 + beta) / 0.16666666666666666d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.3:
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666)
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.3)
		tmp = Float64(1.0 / Float64(Float64(2.0 + beta) / 0.16666666666666666));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.3)
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.3], N[(1.0 / N[(N[(2.0 + beta), $MachinePrecision] / 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.29999999999999982

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

       1. clear-num 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

       2. inv-pow 63.2%

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

   12. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{{\left(\frac{\beta + 2}{0.16666666666666666}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   14. Simplified 63.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   if 5.29999999999999982 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

      1. associate-*l/ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+r+ 86.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. associate-+r+ 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 85.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.3)
   (/ 0.16666666666666666 (+ 2.0 beta))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 0.16666666666666666 / (2.0 + beta);
	} 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 <= 5.3d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + beta)
    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 <= 5.3) {
		tmp = 0.16666666666666666 / (2.0 + beta);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.3:
		tmp = 0.16666666666666666 / (2.0 + beta)
	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 <= 5.3)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + beta));
	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 <= 5.3)
		tmp = 0.16666666666666666 / (2.0 + beta);
	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, 5.3], N[(0.16666666666666666 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.29999999999999982

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 5.29999999999999982 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

      1. associate-*l/ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+r+ 86.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. associate-+r+ 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 85.8%

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

   10. Taylor expanded in alpha around 0 73.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.3)
   (/ 1.0 (/ (+ 2.0 beta) 0.16666666666666666))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} 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 <= 5.3d0) then
        tmp = 1.0d0 / ((2.0d0 + beta) / 0.16666666666666666d0)
    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 <= 5.3) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.3:
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666)
	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 <= 5.3)
		tmp = Float64(1.0 / Float64(Float64(2.0 + beta) / 0.16666666666666666));
	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 <= 5.3)
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	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, 5.3], N[(1.0 / N[(N[(2.0 + beta), $MachinePrecision] / 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.29999999999999982

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

       1. clear-num 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

       2. inv-pow 63.2%

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

   12. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{{\left(\frac{\beta + 2}{0.16666666666666666}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   14. Simplified 63.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   if 5.29999999999999982 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

      1. associate-*l/ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+r+ 86.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. associate-+r+ 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around inf 85.8%

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

   10. Taylor expanded in alpha around 0 73.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 1.0 (/ (+ 2.0 beta) 0.16666666666666666))
   (/ (/ 1.0 (+ 2.0 beta)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} else {
		tmp = (1.0 / (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 <= 6.0d0) then
        tmp = 1.0d0 / ((2.0d0 + beta) / 0.16666666666666666d0)
    else
        tmp = (1.0d0 / (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 <= 6.0) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} else {
		tmp = (1.0 / (2.0 + beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666)
	else:
		tmp = (1.0 / (2.0 + beta)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(1.0 / Float64(Float64(2.0 + beta) / 0.16666666666666666));
	else
		tmp = Float64(Float64(1.0 / Float64(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 <= 6.0)
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	else
		tmp = (1.0 / (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, 6.0], N[(1.0 / N[(N[(2.0 + beta), $MachinePrecision] / 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

       1. clear-num 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

       2. inv-pow 63.2%

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

   12. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{{\left(\frac{\beta + 2}{0.16666666666666666}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   14. Simplified 63.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   if 6 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in beta around inf 85.7%

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

      1. un-div-inv 85.8%

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

      2. +-commutative 85.8%

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

      3. +-commutative 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in alpha around 0 74.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2)
   (/ 1.0 (/ (+ 2.0 beta) 0.16666666666666666))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.2d0) then
        tmp = 1.0d0 / ((2.0d0 + beta) / 0.16666666666666666d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2) {
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2:
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2)
		tmp = Float64(1.0 / Float64(Float64(2.0 + beta) / 0.16666666666666666));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.2)
		tmp = 1.0 / ((2.0 + beta) / 0.16666666666666666);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.2], N[(1.0 / N[(N[(2.0 + beta), $MachinePrecision] / 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.1999999999999993

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

       1. clear-num 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

       2. inv-pow 63.2%

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

   12. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{{\left(\frac{\beta + 2}{0.16666666666666666}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 63.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   14. Simplified 63.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta + 2}{0.16666666666666666}}} \]

   if 8.1999999999999993 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in beta around inf 85.7%

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

      1. un-div-inv 85.8%

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

      2. +-commutative 85.8%

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

      3. +-commutative 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in beta around inf 85.6%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) 0.08333333333333333 (/ 0.16666666666666666 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / 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 <= 2.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.16666666666666666d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.16666666666666666 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.16666666666666666 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.16666666666666666 / 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, 2.0], 0.08333333333333333, N[(0.16666666666666666 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   11. Taylor expanded in beta around 0 63.2%

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

   if 2 < beta

   1. Initial program 81.5%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in beta around 0 17.2%

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

      1. +-commutative 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in alpha around 0 7.1%

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

      1. +-commutative 7.1%

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

   10. Simplified 7.1%

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

   11. Taylor expanded in beta around inf 7.1%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.8% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Simplified 95.4%

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

5. Taylor expanded in beta around 0 71.8%

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

   1. +-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in alpha around 0 44.2%

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

   1. +-commutative 44.2%

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

10. Simplified 44.2%

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

11. Final simplification 44.2%

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

Alternative 15: 45.1% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Simplified 95.4%

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

5. Taylor expanded in beta around 0 71.8%

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

   1. +-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in alpha around 0 44.2%

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

   1. +-commutative 44.2%

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

10. Simplified 44.2%

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

11. Taylor expanded in beta around 0 43.2%

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

12. Final simplification 43.2%

    \[\leadsto 0.08333333333333333 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))