Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 16.8s

Alternatives: 16

Speedup: 3.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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: 99.8% accurate, 1.3× speedup

Math

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

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)
    (* (* t_0 (/ 1.0 (+ 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) / ((t_0 * (1.0 / (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) / ((t_0 * (1.0d0 / (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) / ((t_0 * (1.0 / (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) / ((t_0 * (1.0 / (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(Float64(t_0 * Float64(1.0 / 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) / ((t_0 * (1.0 / (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[(N[(t$95$0 * N[(1.0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.9%

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

   1. associate-/l/ 90.8%

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

   2. associate-/r* 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l+ 81.8%

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

   5. associate-+r+ 81.8%

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

   6. *-commutative 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. +-commutative 81.8%

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

   9. *-commutative 81.8%

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

   10. distribute-rgt1-in 81.8%

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

   11. +-commutative 81.8%

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

   12. times-frac 95.2%

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

3. Simplified 95.2%

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

   1. add-sqr-sqrt 95.1%

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

   2. pow2 95.1%

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

   3. associate-+r+ 95.1%

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

   4. associate-/r* 99.7%

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

   5. associate-+r+ 99.7%

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

5. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

   2. add-sqr-sqrt 99.8%

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

   3. div-inv 99.8%

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

7. Applied egg-rr 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. frac-times 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. +-commutative 99.4%

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

   9. metadata-eval 99.4%

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

   10. associate-+r+ 99.4%

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

   11. +-commutative 99.4%

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

9. Applied egg-rr 99.4%

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

    1. div-inv 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + \beta}{t_0} \cdot \frac{\alpha + 1}{t_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

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 4.2e+126)
     (* (/ (+ 1.0 beta) t_0) (/ (+ alpha 1.0) (* t_0 (+ alpha (+ beta 3.0)))))
     (/ (/ (+ alpha 1.0) t_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 <= 4.2e+126) {
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_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 <= 4.2d+126) then
        tmp = ((1.0d0 + beta) / t_0) * ((alpha + 1.0d0) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) / t_0) / ((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 <= 4.2e+126) {
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_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 <= 4.2e+126:
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) / t_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 <= 4.2e+126)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(Float64(alpha + 1.0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_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 <= 4.2e+126)
		tmp = ((1.0 + beta) / t_0) * ((alpha + 1.0) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) / t_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, 4.2e+126], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.1999999999999998e126

   1. Initial program 98.8%

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

      1. associate-/l/ 97.8%

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

      2. associate-/r* 88.5%

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

      3. +-commutative 88.5%

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

      4. associate-+l+ 88.5%

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

      5. associate-+r+ 88.5%

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

      6. *-commutative 88.5%

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

      7. distribute-rgt1-in 88.5%

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

      8. +-commutative 88.5%

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

      9. *-commutative 88.5%

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

      10. distribute-rgt1-in 88.5%

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

      11. +-commutative 88.5%

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

      12. times-frac 98.7%

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

   3. Simplified 98.7%

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

   if 4.1999999999999998e126 < beta

   1. Initial program 71.8%

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

      1. associate-/l/ 66.1%

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

      2. associate-/r* 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+l+ 58.0%

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

      5. associate-+r+ 58.0%

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

      6. *-commutative 58.0%

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

      7. distribute-rgt1-in 58.0%

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

      8. +-commutative 58.0%

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

      9. *-commutative 58.0%

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

      10. distribute-rgt1-in 58.0%

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

      11. +-commutative 58.0%

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

      12. times-frac 82.6%

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

   3. Simplified 82.6%

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

      1. add-sqr-sqrt 82.6%

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

      2. pow2 82.6%

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

      3. associate-+r+ 82.6%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

      2. add-sqr-sqrt 99.9%

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

      3. div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. frac-times 99.0%

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

      6. metadata-eval 99.0%

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

      7. *-un-lft-identity 99.0%

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

      8. +-commutative 99.0%

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

      9. metadata-eval 99.0%

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

      10. associate-+r+ 99.0%

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

      11. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

   10. Taylor expanded in beta around inf 89.5%

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

       1. associate-+r+ 89.5%

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

   12. Simplified 89.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((1.0 + beta) / t_0) / (t_0 / ((alpha + 1.0) / (3.0 + (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
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = ((1.0d0 + beta) / t_0) / (t_0 / ((alpha + 1.0d0) / (3.0d0 + (alpha + beta))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.9%

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

   1. associate-/l/ 90.8%

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

   2. associate-/r* 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l+ 81.8%

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

   5. associate-+r+ 81.8%

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

   6. *-commutative 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. +-commutative 81.8%

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

   9. *-commutative 81.8%

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

   10. distribute-rgt1-in 81.8%

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

   11. +-commutative 81.8%

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

   12. times-frac 95.2%

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

3. Simplified 95.2%

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

   1. associate-*r/ 95.2%

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

   2. +-commutative 95.2%

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

5. Applied egg-rr 95.2%

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

   1. associate-/l* 95.2%

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

   2. +-commutative 95.2%

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

   3. +-commutative 95.2%

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

   4. associate-/l* 99.5%

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

   5. +-commutative 99.5%

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

   6. +-commutative 99.5%

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

   7. +-commutative 99.5%

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

   8. associate-+r+ 99.5%

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

   9. +-commutative 99.5%

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

   10. +-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((alpha + 1.0) / t_0) / ((alpha + (beta + 3.0)) * (t_0 / (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
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = ((alpha + 1.0d0) / t_0) / ((alpha + (beta + 3.0d0)) * (t_0 / (1.0d0 + beta)))
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) / ((alpha + (beta + 3.0)) * (t_0 / (1.0 + beta)));
}

Python

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

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(alpha + Float64(beta + 3.0)) * Float64(t_0 / Float64(1.0 + beta))))
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) / ((alpha + (beta + 3.0)) * (t_0 / (1.0 + beta)));
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[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.9%

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

   1. associate-/l/ 90.8%

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

   2. associate-/r* 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l+ 81.8%

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

   5. associate-+r+ 81.8%

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

   6. *-commutative 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. +-commutative 81.8%

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

   9. *-commutative 81.8%

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

   10. distribute-rgt1-in 81.8%

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

   11. +-commutative 81.8%

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

   12. times-frac 95.2%

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

3. Simplified 95.2%

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

   1. add-sqr-sqrt 95.1%

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

   2. pow2 95.1%

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

   3. associate-+r+ 95.1%

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

   4. associate-/r* 99.7%

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

   5. associate-+r+ 99.7%

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

5. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

   2. add-sqr-sqrt 99.8%

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

   3. div-inv 99.8%

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

7. Applied egg-rr 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. frac-times 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. +-commutative 99.4%

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

   9. metadata-eval 99.4%

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

   10. associate-+r+ 99.4%

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

   11. +-commutative 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 5: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7e+15)
   (* (/ (+ 1.0 beta) (+ beta 2.0)) (/ (/ 1.0 (+ beta 2.0)) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) * ((1.0d0 / (beta + 2.0d0)) / (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) * ((1.0 / (beta + 2.0)) / (beta + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.7e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.7e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 69.6%

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

      1. associate-/r* 69.6%

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

      2. +-commutative 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in alpha around 0 68.6%

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

   if 1.7e15 < beta

   1. Initial program 79.4%

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

   2. Taylor expanded in beta around inf 81.1%

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

   3. Taylor expanded in alpha around 0 81.1%

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

      1. +-commutative 81.1%

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

   5. Simplified 81.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\beta + 2} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5e+15)
   (/ (/ (+ 1.0 beta) (+ 2.0 (+ alpha beta))) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ alpha 1.0) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5e+15) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.5d+15) then
        tmp = ((1.0d0 + beta) / (2.0d0 + (alpha + beta))) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5e+15) {
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.5e+15:
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + Float64(alpha + beta))) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.5e+15)
		tmp = ((1.0 + beta) / (2.0 + (alpha + beta))) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.5e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5e15

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 69.6%

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

      1. associate-/r* 69.6%

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

      2. +-commutative 69.6%

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

   6. Simplified 69.6%

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

      1. expm1-log1p-u 69.6%

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

      2. expm1-udef 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-/l/ 80.0%

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

      5. +-commutative 80.0%

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

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 69.6%

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

      2. expm1-log1p 69.6%

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

      3. associate-*r/ 69.6%

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

      4. *-rgt-identity 69.6%

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

      5. associate-+r+ 69.6%

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

      6. +-commutative 69.6%

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

      7. +-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. +-commutative 69.6%

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

   10. Simplified 69.6%

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

   if 8.5e15 < beta

   1. Initial program 79.4%

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

   2. Taylor expanded in beta around inf 81.1%

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

   3. Taylor expanded in alpha around 0 81.1%

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

      1. +-commutative 81.1%

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

   5. Simplified 81.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 7: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{t_0}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))))
   (if (<= beta 2.2)
     (/ t_0 (* (+ alpha 2.0) (+ alpha 3.0)))
     (/ t_0 (+ (+ beta 4.0) (* alpha 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 2.2) {
		tmp = t_0 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = t_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 + 1.0d0) / (alpha + (beta + 2.0d0))
    if (beta <= 2.2d0) then
        tmp = t_0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = t_0 / ((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 + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 2.2) {
		tmp = t_0 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = t_0 / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0))
	tmp = 0
	if beta <= 2.2:
		tmp = t_0 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = t_0 / ((beta + 4.0) + (alpha * 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0)))
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(t_0 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(t_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 + 1.0) / (alpha + (beta + 2.0));
	tmp = 0.0;
	if (beta <= 2.2)
		tmp = t_0 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = t_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[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.2], N[(t$95$0 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

      1. add-sqr-sqrt 99.5%

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

      2. pow2 99.5%

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

      3. associate-+r+ 99.5%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

      2. add-sqr-sqrt 99.9%

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

      3. div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. frac-times 99.6%

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

      6. metadata-eval 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. +-commutative 99.6%

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

      9. metadata-eval 99.6%

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

      10. associate-+r+ 99.6%

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

      11. +-commutative 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in beta around 0 97.7%

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

   if 2.2000000000000002 < beta

   1. Initial program 79.9%

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

      1. associate-/l/ 74.3%

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

      2. associate-/r* 60.7%

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

      3. +-commutative 60.7%

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

      4. associate-+l+ 60.7%

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

      5. associate-+r+ 60.7%

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

      6. *-commutative 60.7%

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

      7. distribute-rgt1-in 60.7%

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

      8. +-commutative 60.7%

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

      9. *-commutative 60.7%

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

      10. distribute-rgt1-in 60.7%

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

      11. +-commutative 60.7%

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

      12. times-frac 86.9%

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

   3. Simplified 86.9%

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

      1. add-sqr-sqrt 86.7%

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

      2. pow2 86.7%

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

      3. associate-+r+ 86.7%

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

      4. associate-/r* 99.6%

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

      5. associate-+r+ 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow2 99.6%

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

      2. add-sqr-sqrt 99.7%

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

      3. div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. frac-times 99.1%

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

      6. metadata-eval 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. +-commutative 99.1%

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

      9. metadata-eval 99.1%

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

      10. associate-+r+ 99.1%

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

      11. +-commutative 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in beta around inf 81.5%

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

       1. associate-+r+ 81.5%

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

   12. Simplified 81.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

Alternative 8: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = (1.0 + beta) / ((beta + 2.0) * (6.0 + (beta * 5.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+r+ 93.1%

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

      5. +-commutative 93.1%

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

      6. associate-+r+ 93.1%

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

      7. associate-+r+ 93.1%

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

      8. distribute-rgt1-in 93.1%

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

      9. +-commutative 93.1%

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

      10. *-commutative 93.1%

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

      11. distribute-rgt1-in 93.1%

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

      12. +-commutative 93.1%

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

      13. metadata-eval 93.1%

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

      14. associate-+l+ 93.1%

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

      15. *-commutative 93.1%

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

      16. metadata-eval 93.1%

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

      17. associate-+l+ 93.1%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in beta around 0 92.5%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + 2 \cdot \alpha\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   5. Taylor expanded in alpha around 0 68.3%

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

   if 4.5 < beta

   1. Initial program 79.9%

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

   2. Taylor expanded in beta around inf 79.9%

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

   3. Taylor expanded in alpha around 0 79.9%

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

      1. +-commutative 79.9%

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

   5. Simplified 79.9%

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

4. Final simplification 72.3%

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

Alternative 9: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (* (/ 1.0 (+ alpha 2.0)) 0.16666666666666666)
   (/ (/ (+ alpha 1.0) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * 0.16666666666666666d0
    else
        tmp = ((alpha + 1.0d0) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666
	else:
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) * 0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	else
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 69.8%

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

      1. associate-/r* 69.7%

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

      2. +-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in beta around 0 68.2%

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

   8. Taylor expanded in beta around 0 68.2%

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

   if 2.60000000000000009 < beta

   1. Initial program 79.9%

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

   2. Taylor expanded in beta around inf 79.9%

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

   3. Taylor expanded in alpha around 0 79.9%

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

      1. +-commutative 79.9%

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

   5. Simplified 79.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 10: 91.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (* (/ 1.0 (+ alpha 2.0)) 0.16666666666666666)
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * 0.16666666666666666d0
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) * 0.16666666666666666);
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 69.8%

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

      1. associate-/r* 69.7%

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

      2. +-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in beta around 0 68.2%

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

   8. Taylor expanded in beta around 0 68.2%

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

   if 2.60000000000000009 < beta

   1. Initial program 79.9%

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

   2. Taylor expanded in beta around inf 79.9%

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

   3. Taylor expanded in alpha around 0 71.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 11: 92.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (* (/ 1.0 (+ alpha 2.0)) 0.16666666666666666)
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * 0.16666666666666666d0
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) * 0.16666666666666666);
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 69.8%

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

      1. associate-/r* 69.7%

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

      2. +-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in beta around 0 68.2%

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

   8. Taylor expanded in beta around 0 68.2%

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

   if 2.60000000000000009 < beta

   1. Initial program 79.9%

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

   2. Taylor expanded in beta around inf 79.9%

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

   3. Taylor expanded in alpha around 0 71.3%

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

      1. associate-/r* 71.5%

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

      2. +-commutative 71.5%

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

   5. Simplified 71.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 12: 97.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (* (/ 1.0 (+ alpha 2.0)) 0.16666666666666666)
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = (1.0d0 / (alpha + 2.0d0)) * 0.16666666666666666d0
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(Float64(1.0 / Float64(alpha + 2.0)) * 0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 69.8%

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

      1. associate-/r* 69.7%

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

      2. +-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in beta around 0 68.2%

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

   8. Taylor expanded in beta around 0 68.2%

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

   if 4.20000000000000018 < beta

   1. Initial program 79.9%

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

   2. Taylor expanded in beta around inf 79.9%

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

   3. Taylor expanded in beta around inf 79.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{1}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 13: 45.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\alpha + 2} \cdot 0.16666666666666666 \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 / (alpha + 2.0)) * 0.16666666666666666;
}

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 / (alpha + 2.0d0)) * 0.16666666666666666d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return (1.0 / (alpha + 2.0)) * 0.16666666666666666;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (1.0 / (alpha + 2.0)) * 0.16666666666666666

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(1.0 / Float64(alpha + 2.0)) * 0.16666666666666666)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (1.0 / (alpha + 2.0)) * 0.16666666666666666;
end

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. associate-/l/ 90.8%

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

   2. associate-/r* 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l+ 81.8%

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

   5. associate-+r+ 81.8%

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

   6. *-commutative 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. +-commutative 81.8%

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

   9. *-commutative 81.8%

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

   10. distribute-rgt1-in 81.8%

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

   11. +-commutative 81.8%

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

   12. times-frac 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in alpha around 0 71.9%

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

   1. associate-/r* 71.9%

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

   2. +-commutative 71.9%

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

6. Simplified 71.9%

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

7. Taylor expanded in beta around 0 45.9%

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

8. Taylor expanded in beta around 0 46.2%

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

9. Final simplification 46.2%

   \[\leadsto \frac{1}{\alpha + 2} \cdot 0.16666666666666666 \]

Alternative 14: 12.3% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.16666666666666666d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = 0.16666666666666666;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], 0.16666666666666666, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

      1. associate-/l/ 99.6%

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

      2. associate-/r* 93.1%

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

      3. +-commutative 93.1%

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

      4. associate-+l+ 93.1%

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

      5. associate-+r+ 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-rgt1-in 93.1%

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

      8. +-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. distribute-rgt1-in 93.1%

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

      11. +-commutative 93.1%

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

      12. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in alpha around 0 69.8%

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

      1. associate-/r* 69.7%

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

      2. +-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in beta around 0 68.2%

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

   8. Taylor expanded in beta around inf 14.3%

      \[\leadsto \color{blue}{0.16666666666666666} \]

   if 6 < beta

   1. Initial program 79.9%

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

   2. Taylor expanded in beta around inf 79.9%

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

   3. Taylor expanded in alpha around inf 6.6%

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

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 15: 45.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. associate-/l/ 90.8%

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

   2. associate-/r* 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l+ 81.8%

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

   5. associate-+r+ 81.8%

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

   6. *-commutative 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. +-commutative 81.8%

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

   9. *-commutative 81.8%

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

   10. distribute-rgt1-in 81.8%

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

   11. +-commutative 81.8%

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

   12. times-frac 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in alpha around 0 71.9%

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

   1. associate-/r* 71.9%

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

   2. +-commutative 71.9%

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

6. Simplified 71.9%

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

7. Taylor expanded in beta around 0 46.2%

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

8. Final simplification 46.2%

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

Alternative 16: 10.6% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.16666666666666666
end

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.16666666666666666

Derivation

1. Initial program 92.9%

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

   1. associate-/l/ 90.8%

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

   2. associate-/r* 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l+ 81.8%

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

   5. associate-+r+ 81.8%

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

   6. *-commutative 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. +-commutative 81.8%

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

   9. *-commutative 81.8%

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

   10. distribute-rgt1-in 81.8%

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

   11. +-commutative 81.8%

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

   12. times-frac 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in alpha around 0 71.9%

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

   1. associate-/r* 71.9%

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

   2. +-commutative 71.9%

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

6. Simplified 71.9%

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

7. Taylor expanded in beta around 0 45.9%

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

8. Taylor expanded in beta around inf 10.7%

   \[\leadsto \color{blue}{0.16666666666666666} \]

9. Final simplification 10.7%

   \[\leadsto 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023308 
(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)))