Octave 3.8, jcobi/3

Percentage Accurate: 94.0% → 99.7%

Time: 24.2s

Alternatives: 21

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((alpha + 1.0) / t_0) * (1.0 / (t_0 / ((1.0 + beta) / (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 + (beta + 2.0d0)
    code = ((alpha + 1.0d0) / t_0) * (1.0d0 / (t_0 / ((1.0d0 + beta) / (alpha + (beta + 3.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((alpha + 1.0) / t_0) * (1.0 / (t_0 / ((1.0 + beta) / (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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(t$95$0 / N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

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

6. Applied egg-rr 98.0%

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

   1. unpow-1 98.0%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\alpha + 1}} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 9.2e+35)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (*
      (/ 1.0 (/ t_0 (+ alpha 1.0)))
      (/ 1.0 (+ (+ beta 4.0) (* alpha 2.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9.2e+35) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (1.0 / (t_0 / (alpha + 1.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 9.2d+35) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = (1.0d0 / (t_0 / (alpha + 1.0d0))) * (1.0d0 / ((beta + 4.0d0) + (alpha * 2.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9.2e+35) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (1.0 / (t_0 / (alpha + 1.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 9.2e+35:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = (1.0 / (t_0 / (alpha + 1.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 9.2e+35)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_0 / Float64(alpha + 1.0))) * Float64(1.0 / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 9.2e+35)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = (1.0 / (t_0 / (alpha + 1.0))) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.2e+35], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$0 / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.1999999999999993e35

   1. Initial program 99.8%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in alpha around 0 79.6%

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

   6. Taylor expanded in alpha around 0 65.8%

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

   if 9.1999999999999993e35 < beta

   1. Initial program 76.8%

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

   3. Simplified 94.5%

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

      1. clear-num 94.5%

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

      2. inv-pow 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. unpow-1 94.5%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around inf 83.5%

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

       1. associate-+r+ 83.5%

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

   11. Simplified 83.5%

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

       1. clear-num 83.5%

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

       2. inv-pow 83.5%

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

       3. +-commutative 83.5%

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

   13. Applied egg-rr 83.5%

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

       1. unpow-1 83.5%

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

       2. +-commutative 83.5%

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

   15. Simplified 83.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\alpha + 1}} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 1.4× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (/ (+ alpha 1.0) 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 + (beta + 2.0);
	return ((alpha + 1.0) / 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 + (beta + 2.0d0)
    code = ((alpha + 1.0d0) / 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 + (beta + 2.0);
	return ((alpha + 1.0) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
}

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((alpha + 1.0) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (/ (+ alpha 1.0) 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 + (beta + 2.0);
	return ((alpha + 1.0) / 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 + (beta + 2.0d0)
    code = ((alpha + 1.0d0) / 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 + (beta + 2.0);
	return ((alpha + 1.0) / t_0) * (((1.0 + beta) / t_0) / (alpha + (beta + 3.0)));
}

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	return Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(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 + (beta + 2.0);
	tmp = ((alpha + 1.0) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

   1. clear-num 98.0%

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

   2. associate-+r+ 98.0%

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

   3. *-commutative 98.0%

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

   4. frac-times 94.1%

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

   5. *-un-lft-identity 94.1%

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

   6. +-commutative 94.1%

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

   7. *-commutative 94.1%

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

   8. associate-+r+ 94.1%

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

6. Applied egg-rr 94.1%

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

   1. associate-/r* 98.0%

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

   2. associate-/l* 92.5%

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

   3. associate-*l/ 98.0%

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

   4. *-commutative 98.0%

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

   5. times-frac 99.8%

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

   6. associate-/r* 98.0%

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

   7. *-commutative 98.0%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.06e+36)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (* (/ (+ alpha 1.0) t_0) (/ 1.0 (+ (+ beta 4.0) (* alpha 2.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.06e+36) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.06d+36) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / t_0) * (1.0d0 / ((beta + 4.0d0) + (alpha * 2.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.06e+36) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.06e+36:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((alpha + 1.0) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.06e+36)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(1.0 / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.06e+36)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) / t_0) * (1.0 / ((beta + 4.0) + (alpha * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.06e+36], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.06000000000000002e36

   1. Initial program 99.8%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in alpha around 0 79.6%

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

   6. Taylor expanded in alpha around 0 65.8%

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

   if 1.06000000000000002e36 < beta

   1. Initial program 76.8%

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

   3. Simplified 94.5%

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

      1. clear-num 94.5%

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

      2. inv-pow 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. unpow-1 94.5%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around inf 83.5%

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

       1. associate-+r+ 83.5%

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

   11. Simplified 83.5%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+28}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.65e+28)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (* (+ alpha 1.0) (/ 1.0 beta)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.65e+28) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.65d+28) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / 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 + 2.0);
	double tmp;
	if (beta <= 2.65e+28) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.65e+28:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.65e+28)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.65e+28)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) * (1.0 / 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 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.65e+28], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6500000000000002e28

   1. Initial program 99.8%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in alpha around 0 79.6%

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

   6. Taylor expanded in alpha around 0 65.4%

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

   if 2.6500000000000002e28 < beta

   1. Initial program 78.2%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in beta around inf 82.7%

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

      1. associate-*l/ 82.8%

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

   7. Applied egg-rr 82.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+28}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.1)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.1d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.1:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.1)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.1)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.1], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000009

   1. Initial program 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+r+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in beta around 0 80.1%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 80.1%

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

       2. +-commutative 80.1%

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

   12. Simplified 80.1%

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

   if 3.10000000000000009 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

      1. associate-*l/ 78.3%

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

   7. Applied egg-rr 78.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\frac{1}{\alpha + 2}}{\left(2 - \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9)
   (/ (/ 1.0 (+ alpha 2.0)) (* (- 2.0 alpha) (+ alpha 3.0)))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = (1.0 / (alpha + 2.0)) / ((2.0 - alpha) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.9d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) / ((2.0d0 - alpha) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = (1.0 / (alpha + 2.0)) / ((2.0 - alpha) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.9:
		tmp = (1.0 / (alpha + 2.0)) / ((2.0 - alpha) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.9)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) / Float64(Float64(2.0 - alpha) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.9)
		tmp = (1.0 / (alpha + 2.0)) / ((2.0 - alpha) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.9], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 - alpha), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999991

   1. Initial program 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+r+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 80.3%

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

   8. Taylor expanded in beta around 0 77.8%

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

      1. associate-/r* 77.8%

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

      2. mul-1-neg 77.8%

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

      3. unsub-neg 77.8%

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

      4. +-commutative 77.8%

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

   10. Simplified 77.8%

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

   if 3.89999999999999991 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

      1. associate-*l/ 78.3%

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

   7. Applied egg-rr 78.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\frac{1}{\alpha + 2}}{\left(2 - \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{1}{\left(\alpha + 2\right) \cdot \left(\left(2 - \alpha\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (/ 1.0 (* (+ alpha 2.0) (* (- 2.0 alpha) (+ alpha 3.0))))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = 1.0 / ((alpha + 2.0) * ((2.0 - alpha) * (alpha + 3.0)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.8d0) then
        tmp = 1.0d0 / ((alpha + 2.0d0) * ((2.0d0 - alpha) * (alpha + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = 1.0 / ((alpha + 2.0) * ((2.0 - alpha) * (alpha + 3.0)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = 1.0 / ((alpha + 2.0) * ((2.0 - alpha) * (alpha + 3.0)))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8)
		tmp = Float64(1.0 / Float64(Float64(alpha + 2.0) * Float64(Float64(2.0 - alpha) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8)
		tmp = 1.0 / ((alpha + 2.0) * ((2.0 - alpha) * (alpha + 3.0)));
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8], N[(1.0 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(2.0 - alpha), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+r+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 80.3%

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

   8. Taylor expanded in beta around 0 77.8%

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

   if 3.7999999999999998 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

      1. associate-*l/ 78.3%

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

   7. Applied egg-rr 78.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{1}{\left(\alpha + 2\right) \cdot \left(\left(2 - \alpha\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5)
   (/ (/ 0.5 (+ alpha 2.0)) (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (* (+ alpha 1.0) (/ 1.0 beta)) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = (0.5 / (alpha + 2.0)) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d0) then
        tmp = (0.5d0 / (alpha + 2.0d0)) / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / beta)) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = (0.5 / (alpha + 2.0)) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5:
		tmp = (0.5 / (alpha + 2.0)) / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5)
		tmp = Float64(Float64(0.5 / Float64(alpha + 2.0)) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / beta)) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5)
		tmp = (0.5 / (alpha + 2.0)) / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((alpha + 1.0) * (1.0 / beta)) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5], N[(N[(0.5 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5

   1. Initial program 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+r+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in beta around 0 80.9%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{2 + \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   11. Step-by-step derivation

       1. +-commutative 80.9%

          \[\leadsto \frac{\frac{0.5}{\color{blue}{\alpha + 2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   12. Simplified 80.9%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{\alpha + 2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 6.5 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

      1. associate-*l/ 78.3%

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

   7. Applied egg-rr 78.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\frac{\beta}{\alpha + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (* (/ 1.0 beta) (/ 1.0 (/ beta (+ alpha 1.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / beta) * (1.0 / (beta / (alpha + 1.0)));
	}
	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 <= 6.6d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = (1.0d0 / beta) * (1.0d0 / (beta / (alpha + 1.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / beta) * (1.0 / (beta / (alpha + 1.0)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = (1.0 / beta) * (1.0 / (beta / (alpha + 1.0)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(1.0 / Float64(beta / Float64(alpha + 1.0))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = (1.0 / beta) * (1.0 / (beta / (alpha + 1.0)));
	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, 6.6], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(1.0 / N[(beta / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5999999999999996

   1. Initial program 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+r+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in beta around 0 80.1%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 80.1%

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

       2. +-commutative 80.1%

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

   12. Simplified 80.1%

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

   if 6.5999999999999996 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

   6. Taylor expanded in beta around inf 78.0%

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

      1. clear-num 78.0%

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

      2. inv-pow 78.0%

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

   8. Applied egg-rr 78.0%

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

      1. unpow-1 78.0%

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

   10. Simplified 78.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\frac{\beta}{\alpha + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5

   1. Initial program 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+r+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in beta around 0 80.1%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 80.1%

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

       2. +-commutative 80.1%

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

   12. Simplified 80.1%

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

   if 6.5 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

      1. un-div-inv 78.3%

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

   7. Applied egg-rr 78.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha + 1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.2)
   (/ 0.5 (* (+ beta 2.0) (+ beta 3.0)))
   (* (/ 1.0 beta) (/ (+ alpha 1.0) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.2d0) then
        tmp = 0.5d0 / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = (1.0d0 / beta) * ((alpha + 1.0d0) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.2) {
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.2:
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.2)
		tmp = Float64(0.5 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(Float64(alpha + 1.0) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.2)
		tmp = 0.5 / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.2], N[(0.5 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.1999999999999993

   1. Initial program 99.8%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-+r+ 99.5%

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

      3. *-commutative 99.5%

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

      4. frac-times 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-+r+ 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/r* 99.5%

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

      2. associate-/l* 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-commutative 99.5%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around 0 97.1%

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

       1. +-commutative 97.1%

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

   11. Simplified 97.1%

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

   12. Taylor expanded in alpha around 0 62.7%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   13. Step-by-step derivation

       1. +-commutative 62.7%

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

   14. Simplified 62.7%

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

   if 9.1999999999999993 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

   6. Taylor expanded in beta around inf 78.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha + 1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha + 1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (* (/ 1.0 beta) (/ (+ alpha 1.0) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = (1.0d0 / beta) * ((alpha + 1.0d0) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(Float64(alpha + 1.0) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = (1.0 / beta) * ((alpha + 1.0) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

      13. associate-+r+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in beta around 0 80.1%

       \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 80.1%

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

       2. +-commutative 80.1%

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

   12. Simplified 80.1%

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

   if 6.20000000000000018 < beta

   1. Initial program 79.9%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 78.3%

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

   6. Taylor expanded in beta around inf 78.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha + 1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.0056:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.0056)
   (/ 1.0 (* beta (+ beta 2.0)))
   (* (/ 1.0 beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.0056) {
		tmp = 1.0 / (beta * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (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 (alpha <= 0.0056d0) then
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.0056) {
		tmp = 1.0 / (beta * (beta + 2.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 0.0056:
		tmp = 1.0 / (beta * (beta + 2.0))
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.0056)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.0056)
		tmp = 1.0 / (beta * (beta + 2.0));
	else
		tmp = (1.0 / beta) * (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[alpha, 0.0056], N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.00559999999999999994

   1. Initial program 99.8%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around inf 36.1%

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

   6. Taylor expanded in alpha around 0 35.7%

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

   if 0.00559999999999999994 < alpha

   1. Initial program 80.6%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 17.0%

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

   6. Taylor expanded in beta around inf 16.6%

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

   7. Taylor expanded in alpha around inf 16.2%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.0056:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.0056:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.0056)
   (/ 1.0 (* beta (+ beta 4.0)))
   (* (/ 1.0 beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.0056) {
		tmp = 1.0 / (beta * (beta + 4.0));
	} else {
		tmp = (1.0 / beta) * (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 (alpha <= 0.0056d0) then
        tmp = 1.0d0 / (beta * (beta + 4.0d0))
    else
        tmp = (1.0d0 / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.0056) {
		tmp = 1.0 / (beta * (beta + 4.0));
	} else {
		tmp = (1.0 / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 0.0056:
		tmp = 1.0 / (beta * (beta + 4.0))
	else:
		tmp = (1.0 / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.0056)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 4.0)));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.0056)
		tmp = 1.0 / (beta * (beta + 4.0));
	else
		tmp = (1.0 / beta) * (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[alpha, 0.0056], N[(1.0 / N[(beta * N[(beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.00559999999999999994

   1. Initial program 99.8%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around inf 47.1%

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

       1. associate-+r+ 47.1%

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

   11. Simplified 47.1%

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

   12. Taylor expanded in beta around inf 36.2%

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

   13. Taylor expanded in alpha around 0 35.7%

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

   if 0.00559999999999999994 < alpha

   1. Initial program 80.6%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in beta around inf 17.0%

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

   6. Taylor expanded in beta around inf 16.6%

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

   7. Taylor expanded in alpha around inf 16.2%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.0056:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\beta} \cdot \frac{\alpha + 1}{\beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 / beta) * ((alpha + 1.0) / 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 = (1.0d0 / beta) * ((alpha + 1.0d0) / beta)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

5. Taylor expanded in beta around inf 29.3%

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

6. Taylor expanded in beta around inf 29.5%

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

7. Final simplification 29.5%

   \[\leadsto \frac{1}{\beta} \cdot \frac{\alpha + 1}{\beta} \]
8. Add Preprocessing

Alternative 18: 35.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\beta} \cdot \frac{\alpha}{\beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 / beta) * (alpha / 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 = (1.0d0 / beta) * (alpha / beta)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

5. Taylor expanded in beta around inf 29.3%

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

6. Taylor expanded in beta around inf 29.5%

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

7. Taylor expanded in alpha around inf 20.3%

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

8. Final simplification 20.3%

   \[\leadsto \frac{1}{\beta} \cdot \frac{\alpha}{\beta} \]
9. Add Preprocessing

Alternative 19: 53.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\alpha + 1}{\beta \cdot \beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (alpha + 1.0) / (beta * 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 = (alpha + 1.0d0) / (beta * beta)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

5. Taylor expanded in beta around inf 29.3%

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

   1. expm1-log1p-u 28.8%

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

   2. expm1-udef 22.1%

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

   3. un-div-inv 22.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\beta}}\right)} - 1 \]

7. Applied egg-rr 22.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\beta}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 28.8%

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

   2. expm1-log1p 29.3%

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

   3. associate-/l/ 31.7%

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

   4. +-commutative 31.7%

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

   5. associate-+r+ 31.7%

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

   6. +-commutative 31.7%

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

   7. +-commutative 31.7%

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

9. Simplified 31.7%

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

10. Taylor expanded in beta around inf 30.9%

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

11. Final simplification 30.9%

    \[\leadsto \frac{\alpha + 1}{\beta \cdot \beta} \]
12. Add Preprocessing

Alternative 20: 2.5% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{\alpha} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.5 alpha))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / alpha;
}

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 = 0.5d0 / alpha
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / alpha;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / alpha

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / alpha)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / alpha;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.5 / alpha), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

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

6. Applied egg-rr 98.0%

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

   1. unpow-1 98.0%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in beta around inf 37.6%

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

    1. associate-+r+ 37.6%

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

11. Simplified 37.6%

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

12. Taylor expanded in alpha around inf 4.7%

    \[\leadsto \color{blue}{\frac{0.5}{\alpha}} \]

13. Final simplification 4.7%

    \[\leadsto \frac{0.5}{\alpha} \]
14. Add Preprocessing

Alternative 21: 6.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.5 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / 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 = 0.5d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.5 / beta), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

3. Simplified 98.0%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

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

6. Applied egg-rr 98.0%

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

   1. unpow-1 98.0%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in beta around inf 37.6%

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

    1. associate-+r+ 37.6%

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

11. Simplified 37.6%

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

12. Taylor expanded in beta around inf 29.2%

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

13. Taylor expanded in alpha around inf 4.5%

    \[\leadsto \color{blue}{\frac{0.5}{\beta}} \]

14. Final simplification 4.5%

    \[\leadsto \frac{0.5}{\beta} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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)))