Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.8%

Time: 19.1s

Alternatives: 14

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Simplified 96.4%

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

   1. *-commutative 96.4%

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

   2. clear-num 96.3%

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

   3. clear-num 96.3%

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

   4. frac-times 96.4%

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

   5. metadata-eval 96.4%

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

   6. +-commutative 96.4%

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

6. Applied egg-rr 96.4%

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

   1. associate-/l* 99.0%

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

   2. associate-+r+ 99.0%

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

   3. +-commutative 99.0%

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

   4. +-commutative 99.0%

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

   5. +-commutative 99.0%

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

   6. +-commutative 99.0%

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

   7. +-commutative 99.0%

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

   8. associate-+r+ 99.0%

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

   9. +-commutative 99.0%

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

   10. +-commutative 99.0%

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

   11. +-commutative 99.0%

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

8. Simplified 99.0%

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

   1. inv-pow 99.0%

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

   2. *-commutative 99.0%

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

   3. unpow-prod-down 99.5%

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

   4. inv-pow 99.5%

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

   5. +-commutative 99.5%

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

   6. associate-+r+ 99.5%

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

   7. +-commutative 99.5%

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

   8. clear-num 99.5%

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

   9. +-commutative 99.5%

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

   10. associate-+r+ 99.5%

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

   11. +-commutative 99.5%

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

   12. inv-pow 99.5%

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

   13. associate-/r/ 99.5%

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

   14. associate-+r+ 99.5%

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

10. Applied egg-rr 99.5%

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

    1. associate-*r/ 99.6%

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

    2. *-rgt-identity 99.6%

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

    3. +-commutative 99.6%

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

    4. *-commutative 99.6%

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

    5. +-commutative 99.6%

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

    6. +-commutative 99.6%

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

    7. +-commutative 99.6%

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

12. Simplified 99.6%

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

    1. div-inv 99.6%

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

    2. +-commutative 99.6%

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

14. Applied egg-rr 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Simplified 96.4%

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

   1. *-commutative 96.4%

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

   2. clear-num 96.3%

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

   3. clear-num 96.3%

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

   4. frac-times 96.4%

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

   5. metadata-eval 96.4%

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

   6. +-commutative 96.4%

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

6. Applied egg-rr 96.4%

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

   1. associate-/l* 99.0%

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

   2. associate-+r+ 99.0%

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

   3. +-commutative 99.0%

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

   4. +-commutative 99.0%

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

   5. +-commutative 99.0%

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

   6. +-commutative 99.0%

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

   7. +-commutative 99.0%

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

   8. associate-+r+ 99.0%

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

   9. +-commutative 99.0%

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

   10. +-commutative 99.0%

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

   11. +-commutative 99.0%

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

8. Simplified 99.0%

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

   1. inv-pow 99.0%

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

   2. *-commutative 99.0%

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

   3. unpow-prod-down 99.5%

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

   4. inv-pow 99.5%

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

   5. +-commutative 99.5%

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

   6. associate-+r+ 99.5%

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

   7. +-commutative 99.5%

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

   8. clear-num 99.5%

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

   9. +-commutative 99.5%

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

   10. associate-+r+ 99.5%

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

   11. +-commutative 99.5%

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

   12. inv-pow 99.5%

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

   13. associate-/r/ 99.5%

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

   14. associate-+r+ 99.5%

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

10. Applied egg-rr 99.5%

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

    1. associate-*r/ 99.6%

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

    2. *-rgt-identity 99.6%

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

    3. +-commutative 99.6%

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

    4. *-commutative 99.6%

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

    5. +-commutative 99.6%

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

    6. +-commutative 99.6%

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

    7. +-commutative 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 3: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.8e+52)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (/
    (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta)))
    (+ (+ beta 4.0) (* alpha 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8e+52) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.8d+52) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / ((beta + 4.0d0) + (alpha * 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.8e+52) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / ((beta + 4.0) + (alpha * 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.8e+52:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / ((beta + 4.0) + (alpha * 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.8e+52)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.8e+52)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / ((beta + 4.0) + (alpha * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.8e+52], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.8e52

   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. +-commutative 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 85.4%

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

   6. Taylor expanded in alpha around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. +-commutative 68.4%

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

   8. Simplified 68.4%

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

   if 6.8e52 < beta

   1. Initial program 82.8%

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

   3. Simplified 89.7%

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

      1. *-commutative 89.7%

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

      2. clear-num 89.6%

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

      3. clear-num 89.6%

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

      4. frac-times 89.7%

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

      5. metadata-eval 89.7%

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

      6. +-commutative 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. associate-/l* 97.7%

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

      2. associate-+r+ 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

      7. +-commutative 97.7%

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

      8. associate-+r+ 97.7%

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

      9. +-commutative 97.7%

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

      10. +-commutative 97.7%

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

      11. +-commutative 97.7%

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

   8. Simplified 97.7%

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

      1. inv-pow 97.7%

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

      2. *-commutative 97.7%

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

      3. unpow-prod-down 99.6%

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

      4. inv-pow 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. clear-num 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-+r+ 99.6%

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

      11. +-commutative 99.6%

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

      12. inv-pow 99.6%

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

      13. associate-/r/ 99.6%

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

      14. associate-+r+ 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*r/ 99.7%

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

       2. *-rgt-identity 99.7%

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

       3. +-commutative 99.7%

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

       4. *-commutative 99.7%

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

       5. +-commutative 99.7%

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

       6. +-commutative 99.7%

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

       7. +-commutative 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in beta around inf 87.4%

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

       1. associate-+r+ 87.4%

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

   15. Simplified 87.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.4%

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

3. Simplified 96.4%

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

   1. *-commutative 96.4%

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

   2. clear-num 96.3%

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

   3. clear-num 96.3%

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

   4. frac-times 96.4%

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

   5. metadata-eval 96.4%

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

   6. +-commutative 96.4%

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

6. Applied egg-rr 96.4%

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

   1. associate-/l* 99.0%

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

   2. associate-+r+ 99.0%

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

   3. +-commutative 99.0%

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

   4. +-commutative 99.0%

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

   5. +-commutative 99.0%

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

   6. +-commutative 99.0%

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

   7. +-commutative 99.0%

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

   8. associate-+r+ 99.0%

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

   9. +-commutative 99.0%

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

   10. +-commutative 99.0%

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

   11. +-commutative 99.0%

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

8. Simplified 99.0%

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

   1. inv-pow 99.0%

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

   2. *-commutative 99.0%

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

   3. unpow-prod-down 99.5%

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

   4. inv-pow 99.5%

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

   5. +-commutative 99.5%

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

   6. associate-+r+ 99.5%

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

   7. +-commutative 99.5%

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

   8. clear-num 99.5%

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

   9. +-commutative 99.5%

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

   10. associate-+r+ 99.5%

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

   11. +-commutative 99.5%

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

   12. inv-pow 99.5%

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

   13. associate-/r/ 99.5%

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

   14. associate-+r+ 99.5%

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

10. Applied egg-rr 99.5%

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

    1. associate-*r/ 99.6%

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

    2. *-rgt-identity 99.6%

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

    3. +-commutative 99.6%

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

    4. *-commutative 99.6%

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

    5. +-commutative 99.6%

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

    6. +-commutative 99.6%

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

    7. +-commutative 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in alpha around 0 72.4%

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

    1. associate-/l* 74.1%

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

    2. +-commutative 74.1%

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

    3. +-commutative 74.1%

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

15. Simplified 74.1%

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

16. Final simplification 74.1%

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

Alternative 5: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6e+52)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6e+52) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.6d+52) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (alpha + beta))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6e+52) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6e+52:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6e+52)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.6e+52)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.6e+52], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999999e52

   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. +-commutative 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 85.4%

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

   6. Taylor expanded in alpha around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. +-commutative 68.4%

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

   8. Simplified 68.4%

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

   if 7.5999999999999999e52 < beta

   1. Initial program 82.8%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in beta around inf 86.5%

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

      1. un-div-inv 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-+r+ 86.6%

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

      4. +-commutative 86.6%

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

   7. Applied egg-rr 86.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4.9:\\ \;\;\;\;\frac{0.25}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 4.9) (/ 0.25 (+ 1.0 t_0)) (/ (/ (+ 1.0 alpha) t_0) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 4.9) {
		tmp = 0.25 / (1.0 + t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 4.9d0) then
        tmp = 0.25d0 / (1.0d0 + t_0)
    else
        tmp = ((1.0d0 + alpha) / t_0) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 4.9) {
		tmp = 0.25 / (1.0 + t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 4.9:
		tmp = 0.25 / (1.0 + t_0)
	else:
		tmp = ((1.0 + alpha) / t_0) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 4.9)
		tmp = Float64(0.25 / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 4.9)
		tmp = 0.25 / (1.0 + t_0);
	else
		tmp = ((1.0 + alpha) / t_0) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.9], N[(0.25 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9000000000000004

   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 0 98.3%

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

   4. Taylor expanded in alpha around 0 68.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.9000000000000004 < beta

   1. Initial program 85.3%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.7%

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

      1. un-div-inv 82.8%

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

      2. +-commutative 82.8%

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

      3. associate-+r+ 82.8%

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

      4. +-commutative 82.8%

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

   7. Applied egg-rr 82.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.9:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.1:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.1)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.1) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.1d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.1) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.1:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.1)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.1)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.1], N[(0.25 / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.0999999999999996

   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 0 98.3%

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

   4. Taylor expanded in alpha around 0 68.8%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 7.0999999999999996 < beta

   1. Initial program 85.3%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.7%

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

   6. Taylor expanded in beta around inf 82.5%

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

      1. un-div-inv 82.6%

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

      2. +-commutative 82.6%

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

   8. Applied egg-rr 82.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.1:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6) (/ 0.25 (+ beta 3.0)) (/ 1.0 (* beta (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = 1.0 / (beta * (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5999999999999996

   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 0 98.3%

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

   4. Taylor expanded in alpha around 0 67.4%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 67.4%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4.5999999999999996 < beta

   1. Initial program 85.3%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.7%

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

   6. Taylor expanded in alpha around 0 75.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{2 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6) (/ 0.25 (+ beta 3.0)) (/ (/ 1.0 beta) (+ 2.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (2.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = (1.0d0 / beta) / (2.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (2.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = (1.0 / beta) / (2.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(2.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = (1.0 / beta) / (2.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5999999999999996

   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 0 98.3%

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

   4. Taylor expanded in alpha around 0 67.4%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 67.4%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4.5999999999999996 < beta

   1. Initial program 85.3%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.7%

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

   6. Taylor expanded in alpha around 0 75.9%

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

      1. associate-/r* 76.9%

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

      2. +-commutative 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{2 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5) (/ 0.25 (+ beta 3.0)) (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 98.3%

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

   4. Taylor expanded in alpha around 0 67.4%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 67.4%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 6.5 < beta

   1. Initial program 85.3%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.7%

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

   6. Taylor expanded in beta around inf 82.5%

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

      1. un-div-inv 82.6%

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

      2. +-commutative 82.6%

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

   8. Applied egg-rr 82.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 12.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

      1. associate-/l/ 99.5%

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

      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. +-commutative 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in beta around -inf 30.3%

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

   6. Taylor expanded in alpha around 0 14.3%

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

      1. +-commutative 14.3%

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

      2. +-commutative 14.3%

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

   8. Simplified 14.3%

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

   9. Taylor expanded in beta around 0 14.2%

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

   if 6 < beta

   1. Initial program 85.3%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in beta around inf 82.7%

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

   6. Taylor expanded in alpha around inf 6.9%

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

4. Final simplification 11.5%

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

Alternative 12: 47.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in beta around 0 67.4%

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

   1. +-commutative 67.4%

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

7. Simplified 67.4%

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

8. Taylor expanded in alpha around 0 44.8%

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

   1. +-commutative 44.8%

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

10. Simplified 44.8%

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

11. Final simplification 44.8%

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

Alternative 13: 47.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (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 = 0.25d0 / (beta + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in beta around 0 67.5%

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

4. Taylor expanded in alpha around 0 45.0%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
5. Step-by-step derivation

   1. +-commutative 45.0%

      \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

6. Simplified 45.0%

   \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

7. Final simplification 45.0%

   \[\leadsto \frac{0.25}{\beta + 3} \]
8. Add Preprocessing

Alternative 14: 10.8% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-/l/ 93.0%

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

   2. +-commutative 93.0%

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

   3. associate-+l+ 93.0%

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

   4. *-commutative 93.0%

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

   5. metadata-eval 93.0%

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

   6. associate-+l+ 93.0%

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

   7. metadata-eval 93.0%

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

   8. +-commutative 93.0%

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

   9. metadata-eval 93.0%

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

   10. metadata-eval 93.0%

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

   11. associate-+l+ 93.0%

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

3. Simplified 93.0%

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

5. Taylor expanded in beta around -inf 49.3%

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

6. Taylor expanded in alpha around 0 37.2%

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

   1. +-commutative 37.2%

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

   2. +-commutative 37.2%

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

8. Simplified 37.2%

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

9. Taylor expanded in beta around 0 10.4%

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

10. Final simplification 10.4%

    \[\leadsto 0.16666666666666666 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))