Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.8%

Time: 18.2s

Alternatives: 18

Speedup: 3.2×

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: 94.6% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

3. Simplified 97.3%

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

   1. clear-num 97.3%

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

   2. associate-+r+ 97.3%

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

   3. *-commutative 97.3%

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

   4. frac-times 92.7%

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

   5. *-un-lft-identity 92.7%

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

   6. +-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-+r+ 92.7%

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

5. Applied egg-rr 92.7%

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

   1. associate-/r* 97.3%

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

   2. associate-/l* 92.6%

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

   3. associate-*l/ 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.3%

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

   7. *-commutative 97.3%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-+r+ 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow-1 99.7%

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

    2. +-commutative 99.7%

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

11. Simplified 99.7%

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

    1. associate-*l/ 99.8%

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

    2. +-commutative 99.8%

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

    3. frac-times 99.6%

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

    4. *-un-lft-identity 99.6%

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

    5. associate-+r+ 99.6%

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

    6. +-commutative 99.6%

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

    7. associate-+r+ 99.6%

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

    8. +-commutative 99.6%

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

    9. +-commutative 99.6%

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

13. Applied egg-rr 99.6%

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

14. Taylor expanded in alpha around 0 99.6%

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

    1. associate-+r+ 99.6%

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

    2. associate-*r/ 99.6%

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

    3. metadata-eval 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 36000000000:\\ \;\;\;\;\frac{-1 - \alpha}{t_1} \cdot \frac{-1 - \beta}{t_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 36000000000.0)
     (* (/ (- -1.0 alpha) t_1) (/ (- -1.0 beta) (* t_1 t_0)))
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta (* alpha 2.0)))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 36000000000.0) {
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 36000000000.0d0) then
        tmp = (((-1.0d0) - alpha) / t_1) * (((-1.0d0) - beta) / (t_1 * t_0))
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (beta + (alpha * 2.0d0)))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 36000000000.0) {
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 36000000000.0:
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0))
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 36000000000.0)
		tmp = Float64(Float64(Float64(-1.0 - alpha) / t_1) * Float64(Float64(-1.0 - beta) / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + Float64(alpha * 2.0)))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 36000000000.0)
		tmp = ((-1.0 - alpha) / t_1) * ((-1.0 - beta) / (t_1 * t_0));
	else
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 36000000000.0], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(-1.0 - beta), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6e10

   1. Initial program 99.8%

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

   3. Simplified 99.7%

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

   if 3.6e10 < beta

   1. Initial program 81.3%

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

   3. Simplified 92.0%

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

      1. clear-num 92.0%

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

      2. associate-+r+ 92.0%

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

      3. *-commutative 92.0%

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

      4. frac-times 77.3%

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

      5. *-un-lft-identity 77.3%

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

      6. +-commutative 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-+r+ 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. associate-/r* 92.1%

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

      2. associate-/l* 77.0%

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

      3. associate-*l/ 92.1%

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

      4. *-commutative 92.1%

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

      5. times-frac 99.6%

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

      6. associate-/r* 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. unpow-1 99.5%

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

       2. +-commutative 99.5%

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

   11. Simplified 99.5%

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

       1. associate-*l/ 99.7%

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

       2. +-commutative 99.7%

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

       3. frac-times 99.5%

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

       4. *-un-lft-identity 99.5%

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

       5. associate-+r+ 99.5%

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

       6. +-commutative 99.5%

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

       7. associate-+r+ 99.5%

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

       8. +-commutative 99.5%

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

       9. +-commutative 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in beta around inf 87.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 36000000000:\\ \;\;\;\;\frac{-1 - \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{-1 - \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

3. Simplified 97.3%

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

   1. clear-num 97.3%

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

   2. associate-+r+ 97.3%

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

   3. *-commutative 97.3%

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

   4. frac-times 92.7%

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

   5. *-un-lft-identity 92.7%

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

   6. +-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-+r+ 92.7%

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

5. Applied egg-rr 92.7%

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

   1. associate-/r* 97.3%

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

   2. associate-/l* 92.6%

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

   3. associate-*l/ 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.3%

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

   7. *-commutative 97.3%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

3. Simplified 97.3%

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

   1. clear-num 97.3%

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

   2. associate-+r+ 97.3%

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

   3. *-commutative 97.3%

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

   4. frac-times 92.7%

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

   5. *-un-lft-identity 92.7%

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

   6. +-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-+r+ 92.7%

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

5. Applied egg-rr 92.7%

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

   1. associate-/r* 97.3%

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

   2. associate-/l* 92.6%

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

   3. associate-*l/ 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 99.8%

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

   6. associate-/r* 97.3%

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

   7. *-commutative 97.3%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-+r+ 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow-1 99.7%

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

    2. +-commutative 99.7%

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

11. Simplified 99.7%

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

    1. associate-*l/ 99.8%

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

    2. +-commutative 99.8%

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

    3. frac-times 99.6%

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

    4. *-un-lft-identity 99.6%

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

    5. associate-+r+ 99.6%

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

    6. +-commutative 99.6%

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

    7. associate-+r+ 99.6%

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

    8. +-commutative 99.6%

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

    9. +-commutative 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

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

Alternative 5: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{t_0} \cdot \frac{1}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 2000000000.0)
     (* (/ (/ (+ 1.0 beta) (+ alpha (+ 2.0 beta))) t_0) (/ 1.0 (+ 2.0 beta)))
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta (* alpha 2.0)))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2000000000.0) {
		tmp = (((1.0 + beta) / (alpha + (2.0 + beta))) / t_0) * (1.0 / (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (beta <= 2000000000.0d0) then
        tmp = (((1.0d0 + beta) / (alpha + (2.0d0 + beta))) / t_0) * (1.0d0 / (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (beta + (alpha * 2.0d0)))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2000000000.0) {
		tmp = (((1.0 + beta) / (alpha + (2.0 + beta))) / t_0) * (1.0 / (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 2000000000.0:
		tmp = (((1.0 + beta) / (alpha + (2.0 + beta))) / t_0) * (1.0 / (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 2000000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(alpha + Float64(2.0 + beta))) / t_0) * Float64(1.0 / Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + Float64(alpha * 2.0)))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 2000000000.0)
		tmp = (((1.0 + beta) / (alpha + (2.0 + beta))) / t_0) * (1.0 / (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2000000000.0], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e9

   1. Initial program 99.8%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-+r+ 99.7%

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

      3. *-commutative 99.7%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. times-frac 99.8%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in alpha around 0 82.6%

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

   if 2e9 < beta

   1. Initial program 81.3%

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

   3. Simplified 92.0%

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

      1. clear-num 92.0%

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

      2. associate-+r+ 92.0%

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

      3. *-commutative 92.0%

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

      4. frac-times 77.3%

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

      5. *-un-lft-identity 77.3%

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

      6. +-commutative 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-+r+ 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. associate-/r* 92.1%

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

      2. associate-/l* 77.0%

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

      3. associate-*l/ 92.1%

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

      4. *-commutative 92.1%

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

      5. times-frac 99.6%

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

      6. associate-/r* 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. unpow-1 99.5%

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

       2. +-commutative 99.5%

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

   11. Simplified 99.5%

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

       1. associate-*l/ 99.7%

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

       2. +-commutative 99.7%

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

       3. frac-times 99.5%

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

       4. *-un-lft-identity 99.5%

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

       5. associate-+r+ 99.5%

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

       6. +-commutative 99.5%

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

       7. associate-+r+ 99.5%

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

       8. +-commutative 99.5%

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

       9. +-commutative 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in beta around inf 87.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{1}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 34000000:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\alpha + \left(2 + \beta\right)\right) \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 34000000.0)
     (/ (+ 1.0 beta) (* (+ 2.0 beta) (* (+ alpha (+ 2.0 beta)) t_0)))
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta (* alpha 2.0)))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 34000000.0) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((alpha + (2.0 + beta)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (beta <= 34000000.0d0) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * ((alpha + (2.0d0 + beta)) * t_0))
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (beta + (alpha * 2.0d0)))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 34000000.0) {
		tmp = (1.0 + beta) / ((2.0 + beta) * ((alpha + (2.0 + beta)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 34000000.0:
		tmp = (1.0 + beta) / ((2.0 + beta) * ((alpha + (2.0 + beta)) * t_0))
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 34000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(Float64(alpha + Float64(2.0 + beta)) * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + Float64(alpha * 2.0)))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 34000000.0)
		tmp = (1.0 + beta) / ((2.0 + beta) * ((alpha + (2.0 + beta)) * t_0));
	else
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 34000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4e7

   1. Initial program 99.8%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-+r+ 99.7%

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

      3. *-commutative 99.7%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in alpha around 0 82.6%

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

   if 3.4e7 < beta

   1. Initial program 81.3%

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

   3. Simplified 92.0%

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

      1. clear-num 92.0%

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

      2. associate-+r+ 92.0%

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

      3. *-commutative 92.0%

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

      4. frac-times 77.3%

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

      5. *-un-lft-identity 77.3%

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

      6. +-commutative 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-+r+ 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. associate-/r* 92.1%

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

      2. associate-/l* 77.0%

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

      3. associate-*l/ 92.1%

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

      4. *-commutative 92.1%

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

      5. times-frac 99.6%

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

      6. associate-/r* 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. unpow-1 99.5%

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

       2. +-commutative 99.5%

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

   11. Simplified 99.5%

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

       1. associate-*l/ 99.7%

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

       2. +-commutative 99.7%

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

       3. frac-times 99.5%

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

       4. *-un-lft-identity 99.5%

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

       5. associate-+r+ 99.5%

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

       6. +-commutative 99.5%

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

       7. associate-+r+ 99.5%

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

       8. +-commutative 99.5%

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

       9. +-commutative 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in beta around inf 87.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 34000000:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 100000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 100000000.0)
   (/
    (/ (+ 1.0 beta) (+ 2.0 beta))
    (* (+ alpha (+ 2.0 beta)) (+ 3.0 (+ alpha beta))))
   (/
    (/ (+ 1.0 alpha) (+ 3.0 (+ beta (* alpha 2.0))))
    (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 100000000.0) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 100000000.0d0) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((alpha + (2.0d0 + beta)) * (3.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (beta + (alpha * 2.0d0)))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 100000000.0) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 100000000.0:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (3.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 100000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + Float64(alpha * 2.0)))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 100000000.0)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((alpha + (2.0 + beta)) * (3.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / (3.0 + (beta + (alpha * 2.0)))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 100000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e8

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. *-commutative 99.7%

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

      6. metadata-eval 99.7%

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

      7. associate-+l+ 99.7%

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

      8. metadata-eval 99.7%

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

      9. associate-+l+ 99.7%

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

      10. metadata-eval 99.7%

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

      11. metadata-eval 99.7%

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

      12. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 82.6%

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

   if 1e8 < beta

   1. Initial program 81.3%

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

   3. Simplified 92.0%

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

      1. clear-num 92.0%

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

      2. associate-+r+ 92.0%

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

      3. *-commutative 92.0%

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

      4. frac-times 77.3%

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

      5. *-un-lft-identity 77.3%

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

      6. +-commutative 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-+r+ 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. associate-/r* 92.1%

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

      2. associate-/l* 77.0%

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

      3. associate-*l/ 92.1%

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

      4. *-commutative 92.1%

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

      5. times-frac 99.6%

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

      6. associate-/r* 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. unpow-1 99.5%

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

       2. +-commutative 99.5%

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

   11. Simplified 99.5%

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

       1. associate-*l/ 99.7%

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

       2. +-commutative 99.7%

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

       3. frac-times 99.5%

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

       4. *-un-lft-identity 99.5%

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

       5. associate-+r+ 99.5%

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

       6. +-commutative 99.5%

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

       7. associate-+r+ 99.5%

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

       8. +-commutative 99.5%

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

       9. +-commutative 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in beta around inf 87.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 100000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha \cdot 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1

   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. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   if 1 < beta

   1. Initial program 81.8%

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

   3. Simplified 92.2%

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

      1. clear-num 92.2%

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

      2. associate-+r+ 92.2%

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

      3. *-commutative 92.2%

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

      4. frac-times 77.8%

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

      5. *-un-lft-identity 77.8%

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

      6. +-commutative 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-+r+ 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. associate-/r* 92.3%

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

      2. associate-/l* 77.6%

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

      3. associate-*l/ 92.3%

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

      4. *-commutative 92.3%

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

      5. times-frac 99.6%

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

      6. associate-/r* 92.2%

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

      7. *-commutative 92.2%

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

      8. associate-/r* 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. unpow-1 99.5%

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

       2. +-commutative 99.5%

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

   11. Simplified 99.5%

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

       1. associate-*l/ 99.7%

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

       2. +-commutative 99.7%

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

       3. frac-times 99.5%

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

       4. *-un-lft-identity 99.5%

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

       5. associate-+r+ 99.5%

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

       6. +-commutative 99.5%

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

       7. associate-+r+ 99.5%

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

       8. +-commutative 99.5%

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

       9. +-commutative 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in beta around inf 85.1%

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

4. Final simplification 73.5%

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

Alternative 9: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.25}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\beta}\\ \end{array} \end{array} \]

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 5.0) (/ 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 <= 5.0) {
		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 <= 5.0d0) 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 <= 5.0) {
		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 <= 5.0:
		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 <= 5.0)
		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 <= 5.0)
		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, 5.0], 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 < 5

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   if 5 < beta

   1. Initial program 81.8%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in beta around inf 84.0%

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

      1. un-div-inv 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+r+ 84.2%

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

      4. +-commutative 84.2%

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

      5. +-commutative 84.2%

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

   6. Applied egg-rr 84.2%

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

4. Final simplification 73.2%

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

Alternative 10: 97.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 66.1%

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

      1. +-commutative 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in beta around 0 66.1%

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

      1. *-commutative 66.1%

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

   8. Simplified 66.1%

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

   if 2.7999999999999998 < beta

   1. Initial program 81.8%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in beta around inf 84.0%

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

   5. Taylor expanded in beta around inf 83.9%

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

4. Final simplification 71.8%

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

Alternative 11: 97.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.6d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.6)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   if 7.5999999999999996 < beta

   1. Initial program 81.8%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in beta around inf 84.0%

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

   5. Taylor expanded in beta around inf 83.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 12: 97.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.3)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.3d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3) {
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.3:
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.3)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.3)
		tmp = 0.25 / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.3], 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] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.29999999999999982

   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. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 68.0%

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

   if 4.29999999999999982 < beta

   1. Initial program 81.8%

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

   2. Taylor expanded in beta around -inf 84.2%

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

   3. Taylor expanded in alpha around 0 84.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 13: 91.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 66.1%

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

      1. +-commutative 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in beta around 0 66.1%

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

      1. *-commutative 66.1%

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

   8. Simplified 66.1%

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

   if 2.5 < beta

   1. Initial program 81.8%

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

   2. Taylor expanded in beta around -inf 84.2%

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

   3. Taylor expanded in alpha around 0 76.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 14: 47.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.7d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 0.25 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.7], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7000000000000002

   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. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 66.1%

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

      1. +-commutative 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in beta around 0 66.1%

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

      1. *-commutative 66.1%

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

   8. Simplified 66.1%

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

   if 2.7000000000000002 < beta

   1. Initial program 81.8%

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

   2. Taylor expanded in beta around 0 22.7%

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

   3. Taylor expanded in alpha around 0 6.9%

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

      1. +-commutative 6.9%

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

   5. Simplified 6.9%

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

   6. Taylor expanded in beta around inf 6.9%

      \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 15: 47.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0) 0.08333333333333333 (/ 0.25 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.25 / 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 <= 3.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.25d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.25 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.25 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.25 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.25 / 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, 3.0], 0.08333333333333333, N[(0.25 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   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. Taylor expanded in beta around 0 99.1%

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

   3. Taylor expanded in alpha around 0 66.1%

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

      1. +-commutative 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in beta around 0 65.7%

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

   if 3 < beta

   1. Initial program 81.8%

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

   2. Taylor expanded in beta around 0 22.7%

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

   3. Taylor expanded in alpha around 0 6.9%

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

      1. +-commutative 6.9%

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

   5. Simplified 6.9%

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

   6. Taylor expanded in beta around inf 6.9%

      \[\leadsto \color{blue}{\frac{0.25}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 16: 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 94.0%

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

3. Simplified 84.3%

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

4. Taylor expanded in beta around 0 79.8%

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

5. Taylor expanded in beta around 0 71.5%

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

   1. +-commutative 71.5%

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

7. Simplified 71.5%

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

8. Taylor expanded in alpha around 0 46.9%

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

   1. +-commutative 46.9%

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

10. Simplified 46.9%

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

11. Final simplification 46.9%

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

Alternative 17: 47.9% 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 94.0%

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

2. Taylor expanded in beta around 0 74.7%

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

3. Taylor expanded in alpha around 0 47.1%

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

   1. +-commutative 47.1%

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

5. Simplified 47.1%

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

6. Final simplification 47.1%

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

Alternative 18: 45.8% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

2. Taylor expanded in beta around 0 74.7%

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

3. Taylor expanded in alpha around 0 47.1%

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

   1. +-commutative 47.1%

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

5. Simplified 47.1%

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

6. Taylor expanded in beta around 0 46.0%

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

7. Final simplification 46.0%

   \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023333 
(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)))