Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 24.1s

Alternatives: 17

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 96.1%

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

   1. associate-*l/ 96.1%

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

   2. +-commutative 96.1%

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

   3. associate-+r+ 96.1%

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

   4. associate-/r* 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+r+ 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

   1. clear-num 99.2%

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

   2. inv-pow 99.2%

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

   3. associate-+r+ 99.2%

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

   4. +-commutative 99.2%

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

   5. associate-+r+ 99.2%

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

   6. +-commutative 99.2%

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

   7. *-commutative 99.2%

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

   8. +-commutative 99.2%

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

   9. associate-+r+ 99.2%

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

   10. +-commutative 99.2%

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

   11. associate-+r+ 99.2%

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

   12. +-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. unpow-1 99.2%

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

   2. +-commutative 99.2%

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

   3. +-commutative 99.2%

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

   4. +-commutative 99.2%

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

   5. associate-*l/ 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-/l* 99.2%

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

   8. associate-/r/ 99.2%

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

   9. +-commutative 99.2%

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

   10. +-commutative 99.2%

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

   11. +-commutative 99.2%

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

   12. +-commutative 99.2%

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

   13. +-commutative 99.2%

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

   14. +-commutative 99.2%

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

10. Simplified 99.2%

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

    1. expm1-log1p-u 99.2%

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

    2. expm1-udef 70.9%

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

12. Applied egg-rr 70.9%

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

    1. expm1-def 99.7%

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

    2. expm1-log1p 99.7%

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

    3. associate-*l/ 99.7%

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

    4. *-lft-identity 99.7%

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

    5. associate-/r* 99.8%

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

    6. +-commutative 99.8%

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

    7. +-commutative 99.8%

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

    8. +-commutative 99.8%

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

    9. +-commutative 99.8%

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

    10. +-commutative 99.8%

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

    11. +-commutative 99.8%

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

    12. +-commutative 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 50000000000000:\\ \;\;\;\;\frac{1 + \alpha}{t_1} \cdot \frac{1 + \beta}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{t_1} \cdot \frac{\frac{1 + \alpha}{t_0}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 50000000000000.0)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_0 t_1)))
     (* (/ beta t_1) (/ (/ (+ 1.0 alpha) t_0) t_1)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 50000000000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 50000000000000.0d0) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_0 * t_1))
    else
        tmp = (beta / t_1) * (((1.0d0 + alpha) / t_0) / t_1)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 50000000000000.0) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 50000000000000.0:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1))
	else:
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 50000000000000.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(beta / t_1) * Float64(Float64(Float64(1.0 + alpha) / t_0) / t_1));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 50000000000000.0)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	else
		tmp = (beta / t_1) * (((1.0 + alpha) / t_0) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 50000000000000.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta / t$95$1), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e13

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   if 5e13 < beta

   1. Initial program 83.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in beta around inf 60.6%

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

      1. times-frac 87.7%

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

      2. *-commutative 87.7%

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

   7. Applied egg-rr 87.7%

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

      1. +-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. +-commutative 87.7%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 50000000000000:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha + \left(2 + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{t_0} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 2.3e+14)
     (/ (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 2.0 beta))) (+ (+ alpha 2.0) beta))
     (* (/ beta t_0) (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2.3e+14) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta);
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 2.3d+14) then
        tmp = ((1.0d0 + beta) / ((beta + 3.0d0) * (2.0d0 + beta))) / ((alpha + 2.0d0) + beta)
    else
        tmp = (beta / t_0) * (((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 2.3e+14) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta);
	} else {
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 2.3e+14:
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta)
	else:
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 2.3e+14)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))) / Float64(Float64(alpha + 2.0) + beta));
	else
		tmp = Float64(Float64(beta / t_0) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 2.3e+14)
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta);
	else
		tmp = (beta / t_0) * (((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.3e+14], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision], N[(N[(beta / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3e14

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in alpha around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. +-commutative 71.6%

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

   9. Simplified 71.6%

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

   if 2.3e14 < beta

   1. Initial program 83.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in beta around inf 60.6%

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

      1. times-frac 87.7%

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

      2. *-commutative 87.7%

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

   7. Applied egg-rr 87.7%

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

      1. +-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. +-commutative 87.7%

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

      4. associate-/r* 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha + \left(2 + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + 2\right) + \beta\\ \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha 2.0) beta)))
   (if (<= beta 2.85e+17)
     (/ (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 2.0 beta))) t_0)
     (/
      (* (+ 1.0 alpha) (/ (- 1.0 (/ alpha beta)) (+ alpha (+ beta 3.0))))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 2.85e+17) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - (alpha / beta)) / (alpha + (beta + 3.0)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 2.0d0) + beta
    if (beta <= 2.85d+17) then
        tmp = ((1.0d0 + beta) / ((beta + 3.0d0) * (2.0d0 + beta))) / t_0
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 - (alpha / beta)) / (alpha + (beta + 3.0d0)))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 2.85e+17) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - (alpha / beta)) / (alpha + (beta + 3.0)))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + 2.0) + beta
	tmp = 0
	if beta <= 2.85e+17:
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / t_0
	else:
		tmp = ((1.0 + alpha) * ((1.0 - (alpha / beta)) / (alpha + (beta + 3.0)))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 2.0) + beta)
	tmp = 0.0
	if (beta <= 2.85e+17)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 - Float64(alpha / beta)) / Float64(alpha + Float64(beta + 3.0)))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 2.0) + beta;
	tmp = 0.0;
	if (beta <= 2.85e+17)
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / t_0;
	else
		tmp = ((1.0 + alpha) * ((1.0 - (alpha / beta)) / (alpha + (beta + 3.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 2.85e+17], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 - N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.85e17

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in alpha around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. +-commutative 71.6%

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

   9. Simplified 71.6%

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

   if 2.85e17 < beta

   1. Initial program 83.6%

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

   3. Simplified 87.7%

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

      1. associate-*l/ 87.6%

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

      2. +-commutative 87.6%

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

      3. associate-+r+ 87.6%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in beta around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

   9. Simplified 84.6%

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

   10. Taylor expanded in alpha around inf 84.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + 2\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 84.1%

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

   1. frac-times 96.1%

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

   2. *-commutative 96.1%

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

   3. associate-+r+ 96.1%

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

   4. associate-/r* 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+r+ 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-+r+ 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 96.1%

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

   1. associate-*l/ 96.1%

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

   2. +-commutative 96.1%

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

   3. associate-+r+ 96.1%

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

   4. associate-/r* 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+r+ 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(2 + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6e+17)
   (/ (/ (+ 1.0 beta) (* (+ beta 3.0) (+ 2.0 beta))) (+ (+ alpha 2.0) beta))
   (/ (/ (- alpha -1.0) (+ alpha (+ 2.0 beta))) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+17) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta);
	} else {
		tmp = ((alpha - -1.0) / (alpha + (2.0 + 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 <= 6d+17) then
        tmp = ((1.0d0 + beta) / ((beta + 3.0d0) * (2.0d0 + beta))) / ((alpha + 2.0d0) + beta)
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (2.0d0 + 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 <= 6e+17) {
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta);
	} else {
		tmp = ((alpha - -1.0) / (alpha + (2.0 + beta))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6e+17:
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta)
	else:
		tmp = ((alpha - -1.0) / (alpha + (2.0 + beta))) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6e+17)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))) / Float64(Float64(alpha + 2.0) + beta));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(2.0 + 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 <= 6e+17)
		tmp = ((1.0 + beta) / ((beta + 3.0) * (2.0 + beta))) / ((alpha + 2.0) + beta);
	else
		tmp = ((alpha - -1.0) / (alpha + (2.0 + 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, 6e+17], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6e17

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/r* 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. associate-+r+ 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in alpha around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. +-commutative 71.6%

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

   9. Simplified 71.6%

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

   if 6e17 < beta

   1. Initial program 83.6%

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

      1. associate-/l/ 75.8%

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

      2. +-commutative 75.8%

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

      3. associate-+l+ 75.8%

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

      4. *-commutative 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-+l+ 75.8%

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

      7. metadata-eval 75.8%

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

      8. +-commutative 75.8%

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

      9. metadata-eval 75.8%

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

      10. metadata-eval 75.8%

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

      11. associate-+l+ 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in beta around -inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. sub-neg 86.3%

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

      3. mul-1-neg 86.3%

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

      4. distribute-neg-in 86.3%

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

      5. +-commutative 86.3%

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

      6. mul-1-neg 86.3%

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

      7. distribute-lft-in 86.3%

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

      8. metadata-eval 86.3%

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

      9. mul-1-neg 86.3%

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

      10. unsub-neg 86.3%

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

   7. Simplified 86.3%

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

      1. expm1-log1p-u 86.3%

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

      2. expm1-udef 52.1%

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

      3. associate-+r+ 52.1%

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

      4. *-commutative 52.1%

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

      5. +-commutative 52.1%

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

   9. Applied egg-rr 52.1%

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

       1. expm1-def 86.3%

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

       2. expm1-log1p 86.3%

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

       3. associate-/r* 84.4%

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

       4. +-commutative 84.4%

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

       5. +-commutative 84.4%

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

       6. associate-+r+ 84.4%

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

   11. Simplified 84.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}}{\left(\alpha + 2\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(2 + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.65e+16)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ beta 3.0) (+ 2.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.65d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((beta + 3.0d0) * (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.65e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.65e+16:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.65e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.65e+16)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.65e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.65e16

   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/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

      11. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 84.3%

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

   6. Taylor expanded in alpha around 0 70.9%

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

      1. +-commutative 70.9%

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

      2. +-commutative 70.9%

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

   8. Simplified 70.9%

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

   if 2.65e16 < beta

   1. Initial program 83.6%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in beta around inf 83.9%

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

      1. associate-*l/ 84.0%

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

      2. +-commutative 84.0%

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

   7. Applied egg-rr 84.0%

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

      1. associate-*r/ 84.0%

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

      2. *-rgt-identity 84.0%

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

      3. +-commutative 84.0%

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

      4. +-commutative 84.0%

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

      5. +-commutative 84.0%

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

   9. Simplified 84.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+19}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(2 + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+19)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ beta 3.0) (+ 2.0 beta)))
   (/ (/ (- alpha -1.0) (+ alpha (+ 2.0 beta))) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+19) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (2.0 + 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 <= 1d+19) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((beta + 3.0d0) * (2.0d0 + beta))
    else
        tmp = ((alpha - (-1.0d0)) / (alpha + (2.0d0 + 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 <= 1e+19) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / (alpha + (2.0 + beta))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1e+19:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta))
	else:
		tmp = ((alpha - -1.0) / (alpha + (2.0 + beta))) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+19)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(alpha + Float64(2.0 + 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 <= 1e+19)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((beta + 3.0) * (2.0 + beta));
	else
		tmp = ((alpha - -1.0) / (alpha + (2.0 + 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, 1e+19], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e19

   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/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

      11. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 84.3%

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

   6. Taylor expanded in alpha around 0 70.9%

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

      1. +-commutative 70.9%

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

      2. +-commutative 70.9%

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

   8. Simplified 70.9%

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

   if 1e19 < beta

   1. Initial program 83.6%

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

      1. associate-/l/ 75.8%

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

      2. +-commutative 75.8%

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

      3. associate-+l+ 75.8%

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

      4. *-commutative 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-+l+ 75.8%

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

      7. metadata-eval 75.8%

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

      8. +-commutative 75.8%

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

      9. metadata-eval 75.8%

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

      10. metadata-eval 75.8%

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

      11. associate-+l+ 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in beta around -inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. sub-neg 86.3%

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

      3. mul-1-neg 86.3%

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

      4. distribute-neg-in 86.3%

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

      5. +-commutative 86.3%

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

      6. mul-1-neg 86.3%

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

      7. distribute-lft-in 86.3%

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

      8. metadata-eval 86.3%

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

      9. mul-1-neg 86.3%

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

      10. unsub-neg 86.3%

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

   7. Simplified 86.3%

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

      1. expm1-log1p-u 86.3%

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

      2. expm1-udef 52.1%

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

      3. associate-+r+ 52.1%

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

      4. *-commutative 52.1%

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

      5. +-commutative 52.1%

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

   9. Applied egg-rr 52.1%

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

       1. expm1-def 86.3%

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

       2. expm1-log1p 86.3%

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

       3. associate-/r* 84.4%

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

       4. +-commutative 84.4%

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

       5. +-commutative 84.4%

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

       6. associate-+r+ 84.4%

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

   11. Simplified 84.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+19}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\beta + 3\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\alpha + \left(2 + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.3%

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

   6. Taylor expanded in alpha around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. *-commutative 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in beta around 0 70.4%

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

       1. *-commutative 70.4%

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

   11. Simplified 70.4%

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

   if 2.5 < beta

   1. Initial program 84.2%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in beta around inf 82.4%

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

      1. associate-*l/ 82.5%

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

      2. +-commutative 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-rgt-identity 82.5%

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

      3. +-commutative 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.3%

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

   6. Taylor expanded in alpha around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. *-commutative 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in beta around 0 70.4%

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

       1. *-commutative 70.4%

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

   11. Simplified 70.4%

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

   if 2.7999999999999998 < beta

   1. Initial program 84.2%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in beta around inf 82.4%

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

   6. Taylor expanded in beta around inf 82.1%

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

4. Final simplification 74.2%

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

Alternative 12: 91.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.3%

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

   6. Taylor expanded in alpha around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. *-commutative 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in beta around 0 70.4%

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

       1. *-commutative 70.4%

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

   11. Simplified 70.4%

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

   if 2.5 < beta

   1. Initial program 84.2%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in beta around inf 82.4%

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

   6. Taylor expanded in alpha around 0 75.7%

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

      1. +-commutative 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{2 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ 1.0 beta) (+ 2.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (2.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 / beta) / (2.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 / beta) / (2.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = (1.0 / beta) / (2.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(2.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (1.0 / beta) / (2.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.3%

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

   6. Taylor expanded in alpha around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. *-commutative 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in beta around 0 70.4%

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

       1. *-commutative 70.4%

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

   11. Simplified 70.4%

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

   if 2.5 < beta

   1. Initial program 84.2%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in beta around inf 82.4%

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

      1. expm1-log1p-u 82.4%

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

      2. expm1-udef 50.9%

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

      3. un-div-inv 50.9%

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

      4. +-commutative 50.9%

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

   7. Applied egg-rr 50.9%

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

      1. expm1-def 82.5%

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

      2. expm1-log1p 82.5%

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

      3. associate-/l/ 81.2%

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

      4. +-commutative 81.2%

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

      5. +-commutative 81.2%

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

      6. +-commutative 81.2%

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

   9. Simplified 81.2%

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

   10. Taylor expanded in alpha around 0 75.7%

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

       1. associate-/r* 76.9%

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

       2. +-commutative 76.9%

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

   12. Simplified 76.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{2 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.2 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.2d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.2 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.2 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.2 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.2 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.2 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.3%

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

   6. Taylor expanded in alpha around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. *-commutative 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in beta around 0 70.4%

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

       1. *-commutative 70.4%

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

   11. Simplified 70.4%

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

   if 2.5 < beta

   1. Initial program 84.2%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in beta around 0 21.8%

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

   6. Taylor expanded in alpha around 0 43.9%

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

      1. +-commutative 43.9%

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

      2. *-commutative 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in beta around inf 6.8%

      \[\leadsto \color{blue}{\frac{0.2}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 96.1%

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

   1. associate-*l/ 96.1%

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

   2. +-commutative 96.1%

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

   3. associate-+r+ 96.1%

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

   4. associate-/r* 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+r+ 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in beta around 0 73.5%

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

8. Taylor expanded in alpha around 0 50.3%

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

9. Final simplification 50.3%

   \[\leadsto \frac{0.16666666666666666}{\left(\alpha + 2\right) + \beta} \]
10. Add Preprocessing

Alternative 16: 47.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 96.1%

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

   1. associate-*l/ 96.1%

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

   2. +-commutative 96.1%

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

   3. associate-+r+ 96.1%

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

   4. associate-/r* 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+r+ 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in beta around 0 73.5%

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

8. Taylor expanded in alpha around 0 49.7%

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

9. Final simplification 49.7%

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

Alternative 17: 46.0% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Simplified 96.1%

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

5. Taylor expanded in beta around 0 74.5%

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

6. Taylor expanded in alpha around 0 61.9%

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

   1. +-commutative 61.9%

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

   2. *-commutative 61.9%

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

8. Simplified 61.9%

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

9. Taylor expanded in beta around 0 48.7%

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

10. Final simplification 48.7%

    \[\leadsto 0.08333333333333333 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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)))