Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.8%

Time: 23.0s

Alternatives: 18

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 82.7%

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

   1. times-frac 94.7%

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

   2. +-commutative 94.7%

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

6. Applied egg-rr 94.7%

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

   1. *-un-lft-identity 94.7%

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

   2. +-commutative 94.7%

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

   3. associate-+r+ 94.7%

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

   4. +-commutative 94.7%

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

8. Applied egg-rr 94.7%

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

   1. *-lft-identity 94.7%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

10. Simplified 99.8%

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

    1. associate-*r/ 99.8%

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

    2. associate-+l+ 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-+r+ 99.8%

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

    5. +-commutative 99.8%

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

12. Applied egg-rr 99.8%

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

    1. *-commutative 99.8%

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

    2. clear-num 99.8%

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

    3. un-div-inv 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{1 + \beta}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.7e+68)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_1 t_0)))
     (/ (/ (+ 1.0 alpha) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.7e+68) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_1;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 3.7d+68) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_1 * t_0))
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_1
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.7e+68) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * t_0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_1;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.7e+68:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * t_0))
	else:
		tmp = ((1.0 + alpha) / t_0) / t_1
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.7e+68)
		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) / t_0) / t_1);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.7e+68)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_1 * t_0));
	else
		tmp = ((1.0 + alpha) / t_0) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.7e+68], 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] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.69999999999999998e68

   1. Initial program 99.8%

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

   3. Simplified 93.8%

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

      1. times-frac 99.2%

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

      2. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 3.69999999999999998e68 < beta

   1. Initial program 73.9%

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

      1. associate-/l/ 71.3%

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

      2. +-commutative 71.3%

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

      3. associate-+l+ 71.3%

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

      4. *-commutative 71.3%

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

      5. metadata-eval 71.3%

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

      6. associate-+l+ 71.3%

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

      7. metadata-eval 71.3%

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

      8. associate-+l+ 71.3%

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

      9. metadata-eval 71.3%

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

      10. metadata-eval 71.3%

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

      11. associate-+l+ 71.3%

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

   3. Simplified 71.3%

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

      1. clear-num 71.3%

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

      2. inv-pow 71.3%

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

      3. +-commutative 71.3%

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

      4. distribute-rgt1-in 71.3%

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

      5. fma-define 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. unpow-1 71.3%

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

      2. +-commutative 71.3%

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

      3. +-commutative 71.3%

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

      4. +-commutative 71.3%

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

      5. associate-+l+ 71.3%

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

      6. fma-undefine 71.3%

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

      7. +-commutative 71.3%

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

      8. *-commutative 71.3%

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

      9. +-commutative 71.3%

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

      10. associate-+r+ 71.3%

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

      11. distribute-rgt1-in 71.3%

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

      12. +-commutative 71.3%

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

   8. Simplified 71.3%

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

   9. Taylor expanded in beta around inf 81.0%

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

       1. *-un-lft-identity 81.0%

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

       2. associate-/r* 89.7%

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

       3. +-commutative 89.7%

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

       4. associate-+l+ 89.7%

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

   11. Applied egg-rr 89.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.9%

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

3. Simplified 82.7%

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

   1. times-frac 94.7%

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

   2. +-commutative 94.7%

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

6. Applied egg-rr 94.7%

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

   1. *-un-lft-identity 94.7%

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

   2. +-commutative 94.7%

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

   3. associate-+r+ 94.7%

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

   4. +-commutative 94.7%

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

8. Applied egg-rr 94.7%

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

   1. *-lft-identity 94.7%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 82.7%

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

   1. times-frac 94.7%

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

   2. +-commutative 94.7%

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

6. Applied egg-rr 94.7%

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

   1. *-un-lft-identity 94.7%

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

   2. +-commutative 94.7%

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

   3. associate-+r+ 94.7%

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

   4. +-commutative 94.7%

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

8. Applied egg-rr 94.7%

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

   1. *-lft-identity 94.7%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

10. Simplified 99.8%

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

    1. associate-*r/ 99.8%

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

    2. associate-+l+ 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-+r+ 99.8%

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

    5. +-commutative 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 5: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+19}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+19)
     (/ (+ 1.0 beta) (* t_0 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+19) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1d+19) then
        tmp = (1.0d0 + beta) / (t_0 * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+19) {
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1e+19:
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1e+19)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1e+19)
		tmp = (1.0 + beta) / (t_0 * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+19], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e19

   1. Initial program 99.8%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   8. Simplified 62.6%

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

   if 1e19 < beta

   1. Initial program 79.3%

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

      1. associate-/l/ 77.3%

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

      2. +-commutative 77.3%

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

      3. associate-+l+ 77.3%

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

      4. *-commutative 77.3%

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

      5. metadata-eval 77.3%

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

      6. associate-+l+ 77.3%

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

      7. metadata-eval 77.3%

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

      8. associate-+l+ 77.3%

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

      9. metadata-eval 77.3%

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

      10. metadata-eval 77.3%

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

      11. associate-+l+ 77.3%

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

   3. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt1-in 77.2%

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

      5. fma-define 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow-1 77.2%

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

      2. +-commutative 77.2%

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

      3. +-commutative 77.2%

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

      4. +-commutative 77.2%

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

      5. associate-+l+ 77.2%

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

      6. fma-undefine 77.2%

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

      7. +-commutative 77.2%

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

      8. *-commutative 77.2%

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

      9. +-commutative 77.2%

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

      10. associate-+r+ 77.2%

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

      11. distribute-rgt1-in 77.2%

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

      12. +-commutative 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in beta around inf 82.8%

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

       1. *-un-lft-identity 82.8%

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

       2. associate-/r* 88.5%

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

       3. +-commutative 88.5%

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

       4. associate-+l+ 88.5%

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

   11. Applied egg-rr 88.5%

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

4. Final simplification 71.3%

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

Alternative 6: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.65e+19)
     (/ (+ 1.0 beta) (* t_0 (+ 6.0 (* beta (+ beta 5.0)))))
     (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.65e+19) {
		tmp = (1.0 + beta) / (t_0 * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.65d+19) then
        tmp = (1.0d0 + beta) / (t_0 * (6.0d0 + (beta * (beta + 5.0d0))))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.65e+19) {
		tmp = (1.0 + beta) / (t_0 * (6.0 + (beta * (beta + 5.0))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.65e+19:
		tmp = (1.0 + beta) / (t_0 * (6.0 + (beta * (beta + 5.0))))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.65e+19)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(6.0 + Float64(beta * Float64(beta + 5.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.65e+19)
		tmp = (1.0 + beta) / (t_0 * (6.0 + (beta * (beta + 5.0))));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.65e+19], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.65e19

   1. Initial program 99.8%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in beta around 0 62.6%

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

       1. +-commutative 62.6%

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

       2. unpow2 62.6%

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

       3. distribute-rgt-out 62.6%

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

   11. Simplified 62.6%

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

   if 1.65e19 < beta

   1. Initial program 79.3%

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

      1. associate-/l/ 77.3%

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

      2. +-commutative 77.3%

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

      3. associate-+l+ 77.3%

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

      4. *-commutative 77.3%

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

      5. metadata-eval 77.3%

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

      6. associate-+l+ 77.3%

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

      7. metadata-eval 77.3%

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

      8. associate-+l+ 77.3%

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

      9. metadata-eval 77.3%

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

      10. metadata-eval 77.3%

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

      11. associate-+l+ 77.3%

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

   3. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt1-in 77.2%

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

      5. fma-define 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow-1 77.2%

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

      2. +-commutative 77.2%

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

      3. +-commutative 77.2%

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

      4. +-commutative 77.2%

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

      5. associate-+l+ 77.2%

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

      6. fma-undefine 77.2%

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

      7. +-commutative 77.2%

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

      8. *-commutative 77.2%

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

      9. +-commutative 77.2%

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

      10. associate-+r+ 77.2%

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

      11. distribute-rgt1-in 77.2%

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

      12. +-commutative 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in beta around inf 82.8%

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

       1. *-un-lft-identity 82.8%

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

       2. associate-/r* 88.5%

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

       3. +-commutative 88.5%

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

       4. associate-+l+ 88.5%

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

   11. Applied egg-rr 88.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{t\_0 \cdot \left(\beta + 3\right)}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.1e+38)
     (/ (/ (+ 1.0 beta) (* t_0 (+ beta 3.0))) (+ beta 2.0))
     (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.1e+38) {
		tmp = ((1.0 + beta) / (t_0 * (beta + 3.0))) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3.1d+38) then
        tmp = ((1.0d0 + beta) / (t_0 * (beta + 3.0d0))) / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.1e+38) {
		tmp = ((1.0 + beta) / (t_0 * (beta + 3.0))) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.1e+38:
		tmp = ((1.0 + beta) / (t_0 * (beta + 3.0))) / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.1e+38)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(beta + 3.0))) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.1e+38)
		tmp = ((1.0 + beta) / (t_0 * (beta + 3.0))) / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.1e+38], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000018e38

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in alpha around 0 81.3%

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

   6. Taylor expanded in alpha around 0 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

   8. Simplified 63.2%

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

      1. associate-/r* 62.7%

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

      2. *-un-lft-identity 62.7%

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

      3. times-frac 62.7%

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

   10. Applied egg-rr 62.7%

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

       1. associate-*l/ 62.7%

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

       2. *-lft-identity 62.7%

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

       3. associate-/l/ 62.7%

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

   12. Simplified 62.7%

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

   if 3.10000000000000018e38 < beta

   1. Initial program 78.5%

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

      1. associate-/l/ 76.4%

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

      2. +-commutative 76.4%

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

      3. associate-+l+ 76.4%

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

      4. *-commutative 76.4%

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

      5. metadata-eval 76.4%

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

      6. associate-+l+ 76.4%

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

      7. metadata-eval 76.4%

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

      8. associate-+l+ 76.4%

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

      9. metadata-eval 76.4%

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

      10. metadata-eval 76.4%

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

      11. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

      1. clear-num 76.4%

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

      2. inv-pow 76.4%

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

      3. +-commutative 76.4%

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

      4. distribute-rgt1-in 76.4%

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

      5. fma-define 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. unpow-1 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. +-commutative 76.4%

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

      5. associate-+l+ 76.4%

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

      6. fma-undefine 76.4%

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

      7. +-commutative 76.4%

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

      8. *-commutative 76.4%

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

      9. +-commutative 76.4%

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

      10. associate-+r+ 76.4%

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

      11. distribute-rgt1-in 76.4%

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

      12. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in beta around inf 82.2%

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

       1. *-un-lft-identity 82.2%

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

       2. associate-/r* 89.2%

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

       3. +-commutative 89.2%

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

       4. associate-+l+ 89.2%

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

   11. Applied egg-rr 89.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2e+18)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+18) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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 <= 8.2d+18) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e+18) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2e+18:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2e+18)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.2e+18)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 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, 8.2e+18], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2e18

   1. Initial program 99.8%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in alpha around 0 61.5%

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

       1. +-commutative 61.5%

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

   11. Simplified 61.5%

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

   if 8.2e18 < beta

   1. Initial program 79.3%

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

   3. Taylor expanded in beta around inf 88.2%

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

   4. Taylor expanded in alpha around 0 88.2%

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

      1. +-commutative 88.2%

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

      2. associate-+r+ 88.2%

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

      3. +-commutative 88.2%

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

      4. associate-+r+ 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.9e+23)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9e+23) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.9d+23) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.9e+23) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.9e+23:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.9e+23)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.9e+23)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.9e+23], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.90000000000000013e23

   1. Initial program 99.8%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. +-commutative 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in alpha around 0 61.5%

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

       1. +-commutative 61.5%

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

   11. Simplified 61.5%

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

   if 2.90000000000000013e23 < beta

   1. Initial program 79.3%

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

      1. associate-/l/ 77.3%

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

      2. +-commutative 77.3%

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

      3. associate-+l+ 77.3%

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

      4. *-commutative 77.3%

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

      5. metadata-eval 77.3%

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

      6. associate-+l+ 77.3%

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

      7. metadata-eval 77.3%

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

      8. associate-+l+ 77.3%

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

      9. metadata-eval 77.3%

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

      10. metadata-eval 77.3%

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

      11. associate-+l+ 77.3%

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

   3. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt1-in 77.2%

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

      5. fma-define 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow-1 77.2%

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

      2. +-commutative 77.2%

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

      3. +-commutative 77.2%

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

      4. +-commutative 77.2%

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

      5. associate-+l+ 77.2%

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

      6. fma-undefine 77.2%

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

      7. +-commutative 77.2%

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

      8. *-commutative 77.2%

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

      9. +-commutative 77.2%

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

      10. associate-+r+ 77.2%

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

      11. distribute-rgt1-in 77.2%

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

      12. +-commutative 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in beta around inf 82.8%

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

       1. *-un-lft-identity 82.8%

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

       2. associate-/r* 88.5%

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

       3. +-commutative 88.5%

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

       4. associate-+l+ 88.5%

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

   11. Applied egg-rr 88.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (if (<= beta 4.8e+155)
     (/ (+ 1.0 alpha) (* beta (+ beta 3.0)))
     (/ (/ alpha beta) (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 4.8e+155) {
		tmp = (1.0 + alpha) / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else if (beta <= 4.8d+155) then
        tmp = (1.0d0 + alpha) / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 4.8e+155) {
		tmp = (1.0 + alpha) / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	elif beta <= 4.8e+155:
		tmp = (1.0 + alpha) / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	elseif (beta <= 4.8e+155)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	elseif (beta <= 4.8e+155)
		tmp = (1.0 + alpha) / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.8e+155], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

   if 2.2999999999999998 < beta < 4.80000000000000042e155

   1. Initial program 93.7%

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

   3. Taylor expanded in beta around inf 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. associate-/l/ 80.7%

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

      3. metadata-eval 80.7%

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

      4. associate-+l+ 80.7%

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

      5. metadata-eval 80.7%

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

      6. associate-+r+ 80.7%

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

   5. Applied egg-rr 80.7%

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

      1. *-lft-identity 80.7%

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

      2. *-commutative 80.7%

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

      3. +-commutative 80.7%

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

   7. Simplified 80.7%

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

   8. Taylor expanded in alpha around 0 78.5%

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

   if 4.80000000000000042e155 < beta

   1. Initial program 64.0%

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

   3. Taylor expanded in beta around inf 94.0%

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

   4. Taylor expanded in alpha around 0 94.0%

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

   5. Taylor expanded in alpha around 0 93.9%

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

      1. +-commutative 93.9%

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

   7. Simplified 93.9%

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

   8. Taylor expanded in alpha around inf 92.6%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (if (<= beta 1.05e+160)
     (/ (/ 1.0 (+ beta 3.0)) beta)
     (/ (/ alpha beta) (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 1.05e+160) {
		tmp = (1.0 / (beta + 3.0)) / beta;
	} else {
		tmp = (alpha / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.7d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else if (beta <= 1.05d+160) then
        tmp = (1.0d0 / (beta + 3.0d0)) / beta
    else
        tmp = (alpha / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else if (beta <= 1.05e+160) {
		tmp = (1.0 / (beta + 3.0)) / beta;
	} else {
		tmp = (alpha / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.7:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	elif beta <= 1.05e+160:
		tmp = (1.0 / (beta + 3.0)) / beta
	else:
		tmp = (alpha / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	elseif (beta <= 1.05e+160)
		tmp = Float64(Float64(1.0 / Float64(beta + 3.0)) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	elseif (beta <= 1.05e+160)
		tmp = (1.0 / (beta + 3.0)) / beta;
	else
		tmp = (alpha / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.7], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.05e+160], N[(N[(1.0 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7000000000000002

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

   if 2.7000000000000002 < beta < 1.04999999999999998e160

   1. Initial program 93.9%

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

   3. Taylor expanded in beta around inf 79.1%

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

   4. Taylor expanded in alpha around 0 64.3%

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

      1. inv-pow 64.3%

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

      2. unpow-prod-down 65.2%

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

      3. inv-pow 65.2%

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

      4. +-commutative 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. associate-*l/ 65.4%

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

      2. *-lft-identity 65.4%

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

      3. unpow-1 65.4%

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

   8. Simplified 65.4%

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

   if 1.04999999999999998e160 < beta

   1. Initial program 63.1%

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

   3. Taylor expanded in beta around inf 93.8%

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

   4. Taylor expanded in alpha around 0 93.8%

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

   5. Taylor expanded in alpha around 0 93.7%

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

      1. +-commutative 93.7%

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

   7. Simplified 93.7%

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

   8. Taylor expanded in alpha around inf 93.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

   if 2.60000000000000009 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. associate-+r+ 85.6%

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

      3. +-commutative 85.6%

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

      4. associate-+r+ 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 70.0%

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

Alternative 13: 96.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} 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 <= 2.6d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    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 <= 2.6) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} 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 <= 2.6:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	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 <= 2.6)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	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 <= 2.6)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	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, 2.6], N[(0.16666666666666666 / N[(alpha + 2.0), $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 < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

   if 2.60000000000000009 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 85.5%

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

      1. +-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 69.9%

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

Alternative 14: 91.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} 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 = 0.16666666666666666d0 / (alpha + 2.0d0)
    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 = 0.16666666666666666 / (alpha + 2.0);
	} 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 = 0.16666666666666666 / (alpha + 2.0)
	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(0.16666666666666666 / Float64(alpha + 2.0));
	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 = 0.16666666666666666 / (alpha + 2.0);
	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[(0.16666666666666666 / N[(alpha + 2.0), $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 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

   if 2.5 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 70.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} 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 <= 2.3d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    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 <= 2.3) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	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 <= 2.3)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	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, 2.3], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

   if 2.2999999999999998 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 70.5%

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

      1. associate-/r* 71.1%

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

   6. Simplified 71.1%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = (1.0 / (beta + 3.0)) / 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 <= 2.6d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else
        tmp = (1.0d0 / (beta + 3.0d0)) / beta
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in alpha around 0 81.0%

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

   6. Taylor expanded in alpha around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. +-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in beta around 0 61.3%

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

   if 2.60000000000000009 < beta

   1. Initial program 80.3%

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

   3. Taylor expanded in beta around inf 85.6%

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

   4. Taylor expanded in alpha around 0 70.5%

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

      1. inv-pow 70.5%

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

      2. unpow-prod-down 71.0%

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

      3. inv-pow 71.0%

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

      4. +-commutative 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. associate-*l/ 71.1%

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

      2. *-lft-identity 71.1%

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

      3. unpow-1 71.1%

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

   8. Simplified 71.1%

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

4. Final simplification 64.8%

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

Alternative 17: 44.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Simplified 82.7%

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

5. Taylor expanded in alpha around 0 75.7%

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

6. Taylor expanded in alpha around 0 63.5%

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

   1. +-commutative 63.5%

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

   2. +-commutative 63.5%

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

8. Simplified 63.5%

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

9. Taylor expanded in beta around 0 41.4%

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

10. Final simplification 41.4%

    \[\leadsto \frac{0.16666666666666666}{\alpha + 2} \]
11. Add Preprocessing

Alternative 18: 6.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in beta around inf 32.1%

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

4. Taylor expanded in alpha around 0 26.8%

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

5. Taylor expanded in beta around 0 4.2%

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

6. Final simplification 4.2%

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

Reproduce

herbie shell --seed 2024067 
(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)))