Octave 3.8, jcobi/3

Percentage Accurate: 94.9% → 99.8%

Time: 18.3s

Alternatives: 19

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: 94.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

   1. div-inv 94.8%

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

   2. +-commutative 94.8%

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

   3. *-commutative 94.8%

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

   4. associate-+r+ 94.8%

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

   5. +-commutative 94.8%

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

   6. distribute-rgt1-in 94.8%

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

   7. fma-define 94.8%

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

   8. metadata-eval 94.8%

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

   9. associate-+r+ 94.8%

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

   10. metadata-eval 94.8%

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

   11. associate-+r+ 94.8%

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

4. Applied egg-rr 94.8%

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

   1. associate-*r/ 94.8%

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

   2. *-rgt-identity 94.8%

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

   3. +-commutative 94.8%

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

   4. fma-undefine 94.8%

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

   5. +-commutative 94.8%

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

   6. *-commutative 94.8%

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

   7. +-commutative 94.8%

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

   8. associate-+r+ 94.8%

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

   9. distribute-lft1-in 94.8%

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

   10. +-commutative 94.8%

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

   11. +-commutative 94.8%

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

   12. +-commutative 94.8%

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

   13. +-commutative 94.8%

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

   14. +-commutative 94.8%

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

   15. +-commutative 94.8%

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

   16. +-commutative 94.8%

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

   17. +-commutative 94.8%

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

6. Simplified 94.8%

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

   1. clear-num 94.8%

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

   2. inv-pow 94.8%

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

   3. associate-+l+ 94.8%

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

   4. associate-+l+ 94.8%

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

8. Applied egg-rr 94.8%

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

   1. unpow-1 94.8%

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

   2. +-commutative 94.8%

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

   3. associate-*r/ 99.8%

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

   4. +-commutative 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in alpha around 0 99.8%

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

    1. +-commutative 99.8%

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

13. Simplified 99.8%

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

    1. *-un-lft-identity 99.8%

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

    2. associate-/r/ 99.8%

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

    3. associate-+l+ 99.8%

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

    4. associate-*r/ 94.8%

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

    5. associate-+l+ 94.8%

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

    6. +-commutative 94.8%

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

    7. +-commutative 94.8%

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

    8. associate-+l+ 94.8%

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

15. Applied egg-rr 94.8%

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

    1. *-lft-identity 94.8%

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

    2. associate-*l/ 94.8%

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

    3. *-lft-identity 94.8%

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

    4. associate-/l* 99.8%

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

    5. +-commutative 99.8%

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

    6. +-commutative 99.8%

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

    7. +-commutative 99.8%

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

    8. +-commutative 99.8%

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

    9. +-commutative 99.8%

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

17. Simplified 99.8%

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

18. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha + 1}{t\_0}\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+39}:\\ \;\;\;\;t\_1 \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1 - \frac{\alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ (+ alpha 1.0) t_0)))
   (if (<= beta 2e+39)
     (* t_1 (/ (+ beta 1.0) (* (+ alpha (+ beta 3.0)) t_0)))
     (* t_1 (/ (- 1.0 (/ (* alpha 2.0) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + 1.0) / t_0;
	double tmp;
	if (beta <= 2e+39) {
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = t_1 * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (alpha + 1.0d0) / t_0
    if (beta <= 2d+39) then
        tmp = t_1 * ((beta + 1.0d0) / ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = t_1 * ((1.0d0 - ((alpha * 2.0d0) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (alpha + 1.0) / t_0;
	double tmp;
	if (beta <= 2e+39) {
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = t_1 * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (alpha + 1.0) / t_0
	tmp = 0
	if beta <= 2e+39:
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = t_1 * ((1.0 - ((alpha * 2.0) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(alpha + 1.0) / t_0)
	tmp = 0.0
	if (beta <= 2e+39)
		tmp = Float64(t_1 * Float64(Float64(beta + 1.0) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 - Float64(Float64(alpha * 2.0) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (alpha + 1.0) / t_0;
	tmp = 0.0;
	if (beta <= 2e+39)
		tmp = t_1 * ((beta + 1.0) / ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = t_1 * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2e+39], N[(t$95$1 * N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 - N[(N[(alpha * 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999988e39

   1. Initial program 99.3%

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

   3. Simplified 93.1%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   if 1.99999999999999988e39 < beta

   1. Initial program 82.5%

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

   3. Simplified 66.5%

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

      1. times-frac 91.7%

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

      2. +-commutative 91.7%

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

   6. Applied egg-rr 91.7%

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

   7. Taylor expanded in beta around inf 90.3%

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

   8. Taylor expanded in alpha around inf 90.3%

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

      1. associate-*r/ 90.3%

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

   10. Simplified 90.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{\alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \left(-1 - \beta\right)}{t\_0 \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{t\_0} \cdot \frac{1 - \frac{\alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 9.5e+49)
     (/
      (* (- -1.0 alpha) (- -1.0 beta))
      (* t_0 (* (+ alpha (+ beta 3.0)) t_0)))
     (* (/ (+ alpha 1.0) t_0) (/ (- 1.0 (/ (* alpha 2.0) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9.5e+49) {
		tmp = ((-1.0 - alpha) * (-1.0 - beta)) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 9.5d+49) then
        tmp = (((-1.0d0) - alpha) * ((-1.0d0) - beta)) / (t_0 * ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = ((alpha + 1.0d0) / t_0) * ((1.0d0 - ((alpha * 2.0d0) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9.5e+49) {
		tmp = ((-1.0 - alpha) * (-1.0 - beta)) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 9.5e+49:
		tmp = ((-1.0 - alpha) * (-1.0 - beta)) / (t_0 * ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((alpha * 2.0) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 9.5e+49)
		tmp = Float64(Float64(Float64(-1.0 - alpha) * Float64(-1.0 - beta)) / Float64(t_0 * Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) * Float64(Float64(1.0 - Float64(Float64(alpha * 2.0) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 9.5e+49)
		tmp = ((-1.0 - alpha) * (-1.0 - beta)) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = ((alpha + 1.0) / t_0) * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.5e+49], N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 - N[(N[(alpha * 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.49999999999999969e49

   1. Initial program 99.3%

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

   3. Simplified 93.2%

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

   if 9.49999999999999969e49 < beta

   1. Initial program 82.0%

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

   3. Simplified 65.5%

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

      1. times-frac 91.5%

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

      2. +-commutative 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in beta around inf 90.0%

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

   8. Taylor expanded in alpha around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   10. Simplified 90.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \left(-1 - \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{\alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;t\_0 \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1 - \frac{\alpha \cdot 2 + 4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))))
   (if (<= beta 9.5)
     (* t_0 (/ 1.0 (* (+ alpha 2.0) (+ alpha 3.0))))
     (* t_0 (/ (- 1.0 (/ (+ (* alpha 2.0) 4.0) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 9.5) {
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = t_0 * ((1.0 - (((alpha * 2.0) + 4.0) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 1.0d0) / (alpha + (beta + 2.0d0))
    if (beta <= 9.5d0) then
        tmp = t_0 * (1.0d0 / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = t_0 * ((1.0d0 - (((alpha * 2.0d0) + 4.0d0) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 9.5) {
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = t_0 * ((1.0 - (((alpha * 2.0) + 4.0) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0))
	tmp = 0
	if beta <= 9.5:
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = t_0 * ((1.0 - (((alpha * 2.0) + 4.0) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0)))
	tmp = 0.0
	if (beta <= 9.5)
		tmp = Float64(t_0 * Float64(1.0 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - Float64(Float64(Float64(alpha * 2.0) + 4.0) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	tmp = 0.0;
	if (beta <= 9.5)
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = t_0 * ((1.0 - (((alpha * 2.0) + 4.0) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.5], N[(t$95$0 * N[(1.0 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - N[(N[(N[(alpha * 2.0), $MachinePrecision] + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5

   1. Initial program 99.8%

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

   3. Simplified 93.8%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   if 9.5 < beta

   1. Initial program 83.2%

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

   3. Simplified 67.8%

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

      1. times-frac 92.6%

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

      2. +-commutative 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in beta around inf 83.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{\alpha \cdot 2 + 4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;t\_0 \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))))
   (if (<= beta 9.0)
     (* t_0 (/ 1.0 (* (+ alpha 2.0) (+ alpha 3.0))))
     (* t_0 (/ (- 1.0 (/ 4.0 beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 9.0) {
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = t_0 * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 1.0d0) / (alpha + (beta + 2.0d0))
    if (beta <= 9.0d0) then
        tmp = t_0 * (1.0d0 / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = t_0 * ((1.0d0 - (4.0d0 / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 9.0) {
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = t_0 * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0))
	tmp = 0
	if beta <= 9.0:
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = t_0 * ((1.0 - (4.0 / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0)))
	tmp = 0.0
	if (beta <= 9.0)
		tmp = Float64(t_0 * Float64(1.0 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - Float64(4.0 / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	tmp = 0.0;
	if (beta <= 9.0)
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = t_0 * ((1.0 - (4.0 / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.0], N[(t$95$0 * N[(1.0 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - N[(4.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.8%

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

   3. Simplified 93.8%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   if 9 < beta

   1. Initial program 83.2%

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

   3. Simplified 67.8%

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

      1. times-frac 92.6%

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

      2. +-commutative 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in beta around inf 83.7%

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

   8. Taylor expanded in alpha around 0 83.0%

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

      1. associate-*r/ 83.0%

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

      2. metadata-eval 83.0%

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

   10. Simplified 83.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{if}\;\beta \leq 29:\\ \;\;\;\;t\_0 \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1 - \frac{\alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ alpha 1.0) (+ alpha (+ beta 2.0)))))
   (if (<= beta 29.0)
     (* t_0 (/ 1.0 (* (+ alpha 2.0) (+ alpha 3.0))))
     (* t_0 (/ (- 1.0 (/ (* alpha 2.0) beta)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 29.0) {
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = t_0 * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 1.0d0) / (alpha + (beta + 2.0d0))
    if (beta <= 29.0d0) then
        tmp = t_0 * (1.0d0 / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = t_0 * ((1.0d0 - ((alpha * 2.0d0) / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	double tmp;
	if (beta <= 29.0) {
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = t_0 * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0))
	tmp = 0
	if beta <= 29.0:
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = t_0 * ((1.0 - ((alpha * 2.0) / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0)))
	tmp = 0.0
	if (beta <= 29.0)
		tmp = Float64(t_0 * Float64(1.0 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - Float64(Float64(alpha * 2.0) / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 1.0) / (alpha + (beta + 2.0));
	tmp = 0.0;
	if (beta <= 29.0)
		tmp = t_0 * (1.0 / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = t_0 * ((1.0 - ((alpha * 2.0) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 29.0], N[(t$95$0 * N[(1.0 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - N[(N[(alpha * 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 29

   1. Initial program 99.8%

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

   3. Simplified 93.8%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in beta around 0 98.1%

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

      1. +-commutative 98.1%

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

   9. Simplified 98.1%

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

   if 29 < beta

   1. Initial program 83.2%

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

   3. Simplified 67.8%

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

      1. times-frac 92.6%

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

      2. +-commutative 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in beta around inf 83.7%

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

   8. Taylor expanded in alpha around inf 82.8%

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

      1. associate-*r/ 82.8%

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

   10. Simplified 82.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 29:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{\alpha \cdot 2}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 50000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 50000000.0)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (* (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))) (/ (- 1.0 (/ 4.0 beta)) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 50000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 50000000.0d0) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) * ((1.0d0 - (4.0d0 / beta)) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 50000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 50000000.0:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 50000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(4.0 / beta)) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 50000000.0)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) * ((1.0 - (4.0 / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 50000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(4.0 / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e7

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-+l+ 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 85.1%

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

      1. +-commutative 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in alpha around 0 62.3%

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

   if 5e7 < beta

   1. Initial program 83.0%

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

   3. Simplified 68.6%

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

      1. times-frac 92.6%

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

      2. +-commutative 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in beta around inf 84.7%

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

   8. Taylor expanded in alpha around 0 84.1%

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

      1. associate-*r/ 84.1%

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

      2. metadata-eval 84.1%

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

   10. Simplified 84.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 50000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{4}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 27000000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 27000000000.0)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 27000000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 27000000000.0d0) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 27000000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 27000000000.0:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 27000000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 27000000000.0)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 27000000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7e10

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-+l+ 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 84.8%

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

      1. +-commutative 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in alpha around 0 62.1%

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

   if 2.7e10 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf 84.1%

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

      1. *-un-lft-identity 84.1%

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

      2. metadata-eval 84.1%

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

      3. associate-+l+ 84.1%

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

      4. metadata-eval 84.1%

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

      5. associate-+r+ 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. *-lft-identity 84.1%

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

      2. +-commutative 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 27000000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 27000000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 27000000000.0)
   (/ (/ (+ beta 1.0) (+ beta 2.0)) (* (+ beta 3.0) (+ beta 2.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 27000000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 27000000000.0d0) then
        tmp = ((beta + 1.0d0) / (beta + 2.0d0)) / ((beta + 3.0d0) * (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 27000000000.0) {
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 27000000000.0:
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 27000000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 2.0)) / Float64(Float64(beta + 3.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 27000000000.0)
		tmp = ((beta + 1.0) / (beta + 2.0)) / ((beta + 3.0) * (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (2.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 27000000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7e10

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-+l+ 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 84.8%

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

      1. +-commutative 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in alpha around 0 62.1%

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

   if 2.7e10 < beta

   1. Initial program 82.8%

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

      1. associate-/l/ 80.7%

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

      2. +-commutative 80.7%

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

      3. associate-+l+ 80.7%

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

      4. *-commutative 80.7%

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

      5. metadata-eval 80.7%

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

      6. associate-+l+ 80.7%

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

      7. metadata-eval 80.7%

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

      8. associate-+l+ 80.7%

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

      9. metadata-eval 80.7%

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

      10. metadata-eval 80.7%

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

      11. associate-+l+ 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in beta around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. sub-neg 87.4%

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

      3. mul-1-neg 87.4%

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

      4. distribute-neg-in 87.4%

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

      5. +-commutative 87.4%

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

      6. mul-1-neg 87.4%

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

      7. distribute-lft-in 87.4%

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

      8. metadata-eval 87.4%

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

      9. mul-1-neg 87.4%

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

      10. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

      1. *-un-lft-identity 87.4%

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

      2. associate-+l+ 87.4%

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

      3. +-commutative 87.4%

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

   9. Applied egg-rr 87.4%

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

       1. *-lft-identity 87.4%

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

       2. associate-/r* 84.7%

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

       3. distribute-frac-neg 84.7%

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

       4. +-commutative 84.7%

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

       5. +-commutative 84.7%

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

       6. +-commutative 84.7%

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

       7. associate-+r+ 84.7%

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

       8. +-commutative 84.7%

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

       9. associate-+r+ 84.7%

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

       10. +-commutative 84.7%

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

   11. Simplified 84.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 27000000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 15:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 15.0)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 15.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 15.0d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 15.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 15.0:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 15.0)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 15.0)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 15.0], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 15

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-+l+ 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in beta around 0 84.2%

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

      1. +-commutative 84.2%

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

   10. Simplified 84.2%

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

   if 15 < beta

   1. Initial program 83.2%

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

   3. Taylor expanded in beta around inf 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. metadata-eval 82.7%

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

      3. associate-+l+ 82.7%

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

      4. metadata-eval 82.7%

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

      5. associate-+r+ 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. *-lft-identity 82.7%

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

      2. +-commutative 82.7%

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

   7. Simplified 82.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 15:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta} \cdot \frac{-1 - \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (* (/ -1.0 beta) (/ (- -1.0 alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (-1.0 / beta) * ((-1.0 - 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 <= 5.0d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((-1.0d0) / beta) * (((-1.0d0) - alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = (-1.0 / beta) * ((-1.0 - alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = (-1.0 / beta) * ((-1.0 - alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(-1.0 / beta) * Float64(Float64(-1.0 - alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = (-1.0 / beta) * ((-1.0 - 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, 5.0], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / beta), $MachinePrecision] * N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-+l+ 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in beta around 0 84.2%

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

      1. +-commutative 84.2%

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

   10. Simplified 84.2%

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

   if 5 < beta

   1. Initial program 83.2%

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

   3. Simplified 67.8%

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

      1. times-frac 92.6%

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

      2. +-commutative 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in beta around inf 82.6%

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

   8. Taylor expanded in beta around inf 82.4%

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

4. Final simplification 83.6%

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

Alternative 12: 96.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9)
   (/ 0.5 (* (+ alpha 2.0) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.9d0) then
        tmp = 0.5d0 / ((alpha + 2.0d0) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9) {
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.9:
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9)
		tmp = Float64(0.5 / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9)
		tmp = 0.5 / ((alpha + 2.0) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.9], N[(0.5 / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.89999999999999991

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-+l+ 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in beta around 0 84.2%

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

      1. +-commutative 84.2%

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

   10. Simplified 84.2%

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

   if 2.89999999999999991 < beta

   1. Initial program 83.2%

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

   3. Taylor expanded in beta around inf 82.7%

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

   4. Taylor expanded in alpha around 0 82.5%

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

      1. +-commutative 82.5%

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

   6. Simplified 82.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta * (beta + 3.0);
	double tmp;
	if (alpha <= 1.0) {
		tmp = 1.0 / t_0;
	} else {
		tmp = alpha / 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 = beta * (beta + 3.0d0)
    if (alpha <= 1.0d0) then
        tmp = 1.0d0 / t_0
    else
        tmp = alpha / t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 32.9%

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

   4. Taylor expanded in alpha around 0 31.5%

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

   if 1 < alpha

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf 16.1%

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

   4. Taylor expanded in alpha around 0 15.7%

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

      1. +-commutative 15.7%

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

   6. Simplified 15.7%

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

   7. Taylor expanded in alpha around inf 14.7%

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

      1. +-commutative 14.7%

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

   9. Simplified 14.7%

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

4. Final simplification 25.4%

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

Alternative 14: 51.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   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 95.6%

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

      1. times-frac 99.0%

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

      2. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in beta around inf 32.8%

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

   8. Taylor expanded in alpha around 0 31.5%

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

      1. associate-/r* 32.4%

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

      2. +-commutative 32.4%

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

   10. Simplified 32.4%

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

   if 1 < alpha

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf 16.1%

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

   4. Taylor expanded in alpha around 0 15.7%

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

      1. +-commutative 15.7%

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

   6. Simplified 15.7%

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

   7. Taylor expanded in alpha around inf 14.7%

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

      1. +-commutative 14.7%

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

   9. Simplified 14.7%

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

4. Final simplification 25.9%

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

Alternative 15: 52.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 32.9%

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

   4. Taylor expanded in alpha around 0 31.5%

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

      1. *-un-lft-identity 31.5%

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

      2. +-commutative 31.5%

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

   6. Applied egg-rr 31.5%

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

      1. *-lft-identity 31.5%

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

      2. associate-/r* 32.4%

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

   8. Simplified 32.4%

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

   if 1 < alpha

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf 16.1%

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

   4. Taylor expanded in alpha around 0 15.7%

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

      1. +-commutative 15.7%

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

   6. Simplified 15.7%

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

   7. Taylor expanded in alpha around inf 14.7%

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

      1. +-commutative 14.7%

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

   9. Simplified 14.7%

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

4. Final simplification 25.9%

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

Alternative 16: 54.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   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. Add Preprocessing

   3. Taylor expanded in beta around inf 32.9%

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

   4. Taylor expanded in alpha around 0 31.5%

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

      1. *-un-lft-identity 31.5%

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

      2. +-commutative 31.5%

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

   6. Applied egg-rr 31.5%

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

      1. *-lft-identity 31.5%

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

      2. associate-/r* 32.4%

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

   8. Simplified 32.4%

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

   if 1 < alpha

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf 16.1%

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

   4. Taylor expanded in alpha around 0 15.7%

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

      1. +-commutative 15.7%

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

   6. Simplified 15.7%

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

   7. Taylor expanded in alpha around inf 15.7%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(-1.0 / beta), $MachinePrecision] * N[(N[(-1.0 - alpha), $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 86.0%

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

   1. times-frac 97.5%

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

   2. +-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

7. Taylor expanded in beta around inf 26.8%

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

8. Taylor expanded in beta around inf 27.2%

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

9. Final simplification 27.2%

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

Alternative 18: 49.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in beta around inf 26.8%

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

4. Taylor expanded in alpha around 0 24.1%

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

5. Final simplification 24.1%

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

Alternative 19: 6.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 / 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 = 0.3333333333333333d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in beta around inf 26.8%

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

4. Taylor expanded in alpha around 0 24.1%

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

5. Taylor expanded in beta around 0 4.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]

6. Final simplification 4.0%

   \[\leadsto \frac{0.3333333333333333}{\beta} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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)))