Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.8%

Time: 21.7s

Alternatives: 18

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	return (((alpha + 1.0) / t_0) / t_0) * ((beta + 1.0) / (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 + 2.0d0) + beta
    code = (((alpha + 1.0d0) / t_0) / t_0) * ((beta + 1.0d0) / (alpha + (beta + 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = (alpha + 2.0) + beta;
	tmp = (((alpha + 1.0) / t_0) / t_0) * ((beta + 1.0) / (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[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]}, N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.1%

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

3. Simplified 96.4%

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

   1. associate-*r/ 96.4%

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

   2. +-commutative 96.4%

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

   3. associate-+r+ 96.4%

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

   4. +-commutative 96.4%

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

   5. associate-+r+ 96.4%

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

   6. associate-+r+ 96.4%

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

   7. +-commutative 96.4%

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

   8. associate-+r+ 96.4%

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

6. Applied egg-rr 96.4%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + 2\right) + \beta\\ \mathbf{if}\;\beta \leq 530000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{t\_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha 2.0) beta)))
   (if (<= beta 530000000.0)
     (/ (+ beta 1.0) (* (+ alpha (+ 2.0 beta)) (* (+ beta 3.0) (+ 2.0 beta))))
     (* (/ (/ (+ alpha 1.0) t_0) t_0) (- 1.0 (/ (+ alpha 2.0) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 530000000.0) {
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + 2.0d0) + beta
    if (beta <= 530000000.0d0) then
        tmp = (beta + 1.0d0) / ((alpha + (2.0d0 + beta)) * ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = (((alpha + 1.0d0) / t_0) / t_0) * (1.0d0 - ((alpha + 2.0d0) / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + 2.0) + beta;
	double tmp;
	if (beta <= 530000000.0) {
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + 2.0) + beta
	tmp = 0
	if beta <= 530000000.0:
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + 2.0) + beta)
	tmp = 0.0
	if (beta <= 530000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / t_0) / t_0) * Float64(1.0 - Float64(Float64(alpha + 2.0) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + 2.0) + beta;
	tmp = 0.0;
	if (beta <= 530000000.0)
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = (((alpha + 1.0) / t_0) / t_0) * (1.0 - ((alpha + 2.0) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 530000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.3e8

   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 93.2%

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

   5. Taylor expanded in alpha around 0 80.1%

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

   6. Taylor expanded in alpha around 0 64.5%

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

   if 5.3e8 < beta

   1. Initial program 82.2%

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

   3. Simplified 89.6%

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

      1. associate-*r/ 89.6%

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

      2. +-commutative 89.6%

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

      3. associate-+r+ 89.6%

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

      4. +-commutative 89.6%

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

      5. associate-+r+ 89.6%

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

      6. associate-+r+ 89.6%

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

      7. +-commutative 89.6%

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

      8. associate-+r+ 89.6%

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

   6. Applied egg-rr 89.6%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 85.0%

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

       1. mul-1-neg 85.0%

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

       2. unsub-neg 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 530000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\left(\alpha + 2\right) + \beta} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{\alpha + 1}{\beta}}{\left(\alpha + 2\right) + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e+16)
   (/ (+ beta 1.0) (* (+ alpha (+ 2.0 beta)) (* (+ beta 3.0) (+ 2.0 beta))))
   (*
    (/ (+ beta 1.0) (+ alpha (+ beta 3.0)))
    (/ (/ (+ alpha 1.0) beta) (+ (+ alpha 2.0) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+16) {
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * (((alpha + 1.0) / beta) / ((alpha + 2.0) + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d+16) then
        tmp = (beta + 1.0d0) / ((alpha + (2.0d0 + beta)) * ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * (((alpha + 1.0d0) / beta) / ((alpha + 2.0d0) + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+16) {
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * (((alpha + 1.0) / beta) / ((alpha + 2.0) + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.5e+16:
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * (((alpha + 1.0) / beta) / ((alpha + 2.0) + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e+16)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(alpha + 2.0) + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5e+16)
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * (((alpha + 1.0) / beta) / ((alpha + 2.0) + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e16

   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 93.3%

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

   5. Taylor expanded in alpha around 0 80.3%

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

   6. Taylor expanded in alpha around 0 64.4%

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

   if 1.5e16 < beta

   1. Initial program 81.8%

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

   3. Simplified 89.3%

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

      1. associate-*r/ 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+r+ 89.3%

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

      4. +-commutative 89.3%

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

      5. associate-+r+ 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+r+ 89.3%

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

   6. Applied egg-rr 89.3%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 86.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{\alpha + 1}{\beta}}{\left(\alpha + 2\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 96.4%

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

   1. associate-*l/ 96.3%

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

   2. +-commutative 96.3%

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

   3. associate-+r+ 96.3%

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

   4. associate-/r* 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+r+ 99.8%

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

   8. associate-+r+ 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in alpha around 0 71.3%

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

   1. +-commutative 71.3%

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

9. Simplified 71.3%

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

10. Final simplification 71.3%

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

Alternative 5: 97.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 65.0%

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

   if 2 < beta

   1. Initial program 82.6%

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

   3. Simplified 89.8%

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

      1. associate-*l/ 89.8%

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

      2. +-commutative 89.8%

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

      3. associate-+r+ 89.8%

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

      4. associate-/r* 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. associate-+r+ 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf 83.7%

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

4. Final simplification 71.3%

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

Alternative 6: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + 2\right) + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.2e+14)
   (/ (+ beta 1.0) (* (+ alpha (+ 2.0 beta)) (* (+ beta 3.0) (+ 2.0 beta))))
   (/
    (* (+ alpha 1.0) (/ 1.0 (+ alpha (+ beta 3.0))))
    (+ (+ alpha 2.0) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2e+14) {
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / (alpha + (beta + 3.0)))) / ((alpha + 2.0) + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.2d+14) then
        tmp = (beta + 1.0d0) / ((alpha + (2.0d0 + beta)) * ((beta + 3.0d0) * (2.0d0 + beta)))
    else
        tmp = ((alpha + 1.0d0) * (1.0d0 / (alpha + (beta + 3.0d0)))) / ((alpha + 2.0d0) + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2e+14) {
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / (alpha + (beta + 3.0)))) / ((alpha + 2.0) + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.2e+14:
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)))
	else:
		tmp = ((alpha + 1.0) * (1.0 / (alpha + (beta + 3.0)))) / ((alpha + 2.0) + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.2e+14)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(alpha + Float64(2.0 + beta)) * Float64(Float64(beta + 3.0) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 / Float64(alpha + Float64(beta + 3.0)))) / Float64(Float64(alpha + 2.0) + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.2e+14)
		tmp = (beta + 1.0) / ((alpha + (2.0 + beta)) * ((beta + 3.0) * (2.0 + beta)));
	else
		tmp = ((alpha + 1.0) * (1.0 / (alpha + (beta + 3.0)))) / ((alpha + 2.0) + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.2e+14], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.2e14

   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 93.3%

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

   5. Taylor expanded in alpha around 0 80.3%

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

   6. Taylor expanded in alpha around 0 64.4%

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

   if 7.2e14 < beta

   1. Initial program 81.8%

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

   3. Simplified 89.3%

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

      1. associate-*l/ 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+r+ 89.3%

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

      4. associate-/r* 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. associate-+r+ 99.7%

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

      9. associate-+r+ 99.7%

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

      10. +-commutative 99.7%

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

      11. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in beta around inf 86.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + 2\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \left(\alpha + \beta\right)\right) + 1\\ \mathbf{if}\;\beta \leq 4.1:\\ \;\;\;\;\frac{0.25}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 (+ alpha beta)) 1.0)))
   (if (<= beta 4.1) (/ 0.25 t_0) (/ (/ (+ alpha 1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + (alpha + beta)) + 1.0;
	double tmp;
	if (beta <= 4.1) {
		tmp = 0.25 / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / 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 = (2.0d0 + (alpha + beta)) + 1.0d0
    if (beta <= 4.1d0) then
        tmp = 0.25d0 / t_0
    else
        tmp = ((alpha + 1.0d0) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (2.0 + (alpha + beta)) + 1.0;
	double tmp;
	if (beta <= 4.1) {
		tmp = 0.25 / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (2.0 + (alpha + beta)) + 1.0
	tmp = 0
	if beta <= 4.1:
		tmp = 0.25 / t_0
	else:
		tmp = ((alpha + 1.0) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + Float64(alpha + beta)) + 1.0)
	tmp = 0.0
	if (beta <= 4.1)
		tmp = Float64(0.25 / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (2.0 + (alpha + beta)) + 1.0;
	tmp = 0.0;
	if (beta <= 4.1)
		tmp = 0.25 / t_0;
	else
		tmp = ((alpha + 1.0) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[beta, 4.1], N[(0.25 / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 65.0%

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

   if 4.0999999999999996 < beta

   1. Initial program 82.6%

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

   3. Taylor expanded in beta around -inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. mul-1-neg 83.2%

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

      5. distribute-neg-in 83.2%

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

      6. +-commutative 83.2%

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

      7. mul-1-neg 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. metadata-eval 83.2%

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

      10. mul-1-neg 83.2%

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

      11. unsub-neg 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1:\\ \;\;\;\;\frac{0.25}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 0.25 (+ beta 3.0))
   (if (<= beta 2e+156)
     (/ 1.0 (* beta beta))
     (* (/ alpha beta) (/ 1.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else if (beta <= 2e+156) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else if (beta <= 2d+156) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = (alpha / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else if (beta <= 2e+156) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (beta + 3.0)
	elif beta <= 2e+156:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = (alpha / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	elseif (beta <= 2e+156)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (beta + 3.0);
	elseif (beta <= 2e+156)
		tmp = 1.0 / (beta * beta);
	else
		tmp = (alpha / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2e+156], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 63.4%

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

      1. +-commutative 63.4%

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

   6. Simplified 63.4%

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

   if 6 < beta < 2e156

   1. Initial program 94.6%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in beta around inf 78.6%

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

   6. Taylor expanded in beta around inf 78.2%

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

      1. clear-num 78.3%

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

      2. frac-times 76.8%

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

      3. metadata-eval 76.8%

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

      4. +-commutative 76.8%

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

   8. Applied egg-rr 76.8%

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

   9. Taylor expanded in alpha around 0 64.3%

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

   if 2e156 < beta

   1. Initial program 72.2%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in beta around inf 87.0%

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

   6. Taylor expanded in beta around inf 86.9%

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

   7. Taylor expanded in alpha around inf 86.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6)
   (/ 0.25 (+ beta 3.0))
   (if (<= beta 3.5e+155)
     (/ 1.0 (* beta (+ 2.0 beta)))
     (* (/ alpha beta) (/ 1.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else if (beta <= 3.5e+155) {
		tmp = 1.0 / (beta * (2.0 + beta));
	} else {
		tmp = (alpha / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.6d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else if (beta <= 3.5d+155) then
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    else
        tmp = (alpha / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else if (beta <= 3.5e+155) {
		tmp = 1.0 / (beta * (2.0 + beta));
	} else {
		tmp = (alpha / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = 0.25 / (beta + 3.0)
	elif beta <= 3.5e+155:
		tmp = 1.0 / (beta * (2.0 + beta))
	else:
		tmp = (alpha / beta) * (1.0 / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	elseif (beta <= 3.5e+155)
		tmp = Float64(1.0 / Float64(beta * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(alpha / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = 0.25 / (beta + 3.0);
	elseif (beta <= 3.5e+155)
		tmp = 1.0 / (beta * (2.0 + beta));
	else
		tmp = (alpha / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.5e+155], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.5999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 63.4%

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

      1. +-commutative 63.4%

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

   6. Simplified 63.4%

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

   if 4.5999999999999996 < beta < 3.49999999999999985e155

   1. Initial program 94.6%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in beta around inf 78.6%

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

   6. Taylor expanded in alpha around 0 64.4%

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

   if 3.49999999999999985e155 < beta

   1. Initial program 72.2%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in beta around inf 87.0%

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

   6. Taylor expanded in beta around inf 86.9%

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

   7. Taylor expanded in alpha around inf 86.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 4.9000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 65.0%

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

   if 4.9000000000000004 < beta

   1. Initial program 82.6%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around inf 83.1%

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

      1. associate-*l/ 83.1%

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

      2. +-commutative 83.1%

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

      3. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

      1. associate-*r/ 83.2%

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

      2. *-rgt-identity 83.2%

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

      3. associate-+r+ 83.2%

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

      4. +-commutative 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 71.1%

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

Alternative 11: 97.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/ 0.25 (+ (+ 2.0 (+ alpha beta)) 1.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / ((2.0 + (alpha + beta)) + 1.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.4d0) then
        tmp = 0.25d0 / ((2.0d0 + (alpha + beta)) + 1.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / ((2.0 + (alpha + beta)) + 1.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 0.25 / ((2.0 + (alpha + beta)) + 1.0)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(0.25 / Float64(Float64(2.0 + Float64(alpha + beta)) + 1.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 0.25 / ((2.0 + (alpha + beta)) + 1.0);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.4], N[(0.25 / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 65.0%

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

   if 6.4000000000000004 < beta

   1. Initial program 82.6%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around inf 83.1%

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

   6. Taylor expanded in beta around inf 82.9%

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

      1. un-div-inv 82.9%

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

      2. +-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4) (/ 0.25 (+ beta 3.0)) (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.4d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.4], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 63.4%

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

      1. +-commutative 63.4%

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

   6. Simplified 63.4%

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

   if 6.4000000000000004 < beta

   1. Initial program 82.6%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around inf 83.1%

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

   6. Taylor expanded in beta around inf 82.9%

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

      1. un-div-inv 82.9%

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

      2. +-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.92:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.92)
   (+ 0.08333333333333333 (* alpha -0.027777777777777776))
   (/ 0.16666666666666666 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.92) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.16666666666666666 / 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 <= 1.92d0) then
        tmp = 0.08333333333333333d0 + (alpha * (-0.027777777777777776d0))
    else
        tmp = 0.16666666666666666d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.92) {
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	} else {
		tmp = 0.16666666666666666 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.92:
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776)
	else:
		tmp = 0.16666666666666666 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.92)
		tmp = Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776));
	else
		tmp = Float64(0.16666666666666666 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.92)
		tmp = 0.08333333333333333 + (alpha * -0.027777777777777776);
	else
		tmp = 0.16666666666666666 / 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, 1.92], N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9199999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around 0 98.6%

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

      1. +-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. un-div-inv 98.6%

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

      2. +-commutative 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

   9. Applied egg-rr 98.6%

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

   10. Taylor expanded in beta around 0 98.6%

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

   11. Taylor expanded in alpha around 0 62.8%

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

       1. *-commutative 62.8%

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

   13. Simplified 62.8%

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

   if 1.9199999999999999 < beta

   1. Initial program 82.6%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around 0 18.9%

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

      1. +-commutative 18.9%

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

   7. Simplified 18.9%

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

   8. Taylor expanded in alpha around 0 5.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   9. Step-by-step derivation

      1. *-commutative 5.4%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   10. Simplified 5.4%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   11. Taylor expanded in alpha around 0 7.0%

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

       1. +-commutative 7.0%

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

   13. Simplified 7.0%

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

   14. Taylor expanded in beta around inf 7.0%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.92:\\ \;\;\;\;0.08333333333333333 + \alpha \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) (/ 0.25 (+ beta 3.0)) (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 63.4%

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

      1. +-commutative 63.4%

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

   6. Simplified 63.4%

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

   if 6 < beta

   1. Initial program 82.6%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around inf 83.1%

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

   6. Taylor expanded in beta around inf 82.9%

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

      1. clear-num 82.8%

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

      2. frac-times 82.1%

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

      3. metadata-eval 82.1%

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

      4. +-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

   9. Taylor expanded in alpha around 0 75.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) 0.08333333333333333 (/ 0.16666666666666666 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / 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 <= 2.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.16666666666666666d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.16666666666666666 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.16666666666666666 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.16666666666666666 / 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, 2.0], 0.08333333333333333, N[(0.16666666666666666 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around 0 98.6%

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

      1. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   9. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   10. Simplified 62.8%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   11. Taylor expanded in alpha around 0 63.1%

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

       1. +-commutative 63.1%

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

   13. Simplified 63.1%

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

   14. Taylor expanded in beta around 0 63.1%

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

   if 2 < beta

   1. Initial program 82.6%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in beta around 0 18.9%

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

      1. +-commutative 18.9%

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

   7. Simplified 18.9%

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

   8. Taylor expanded in alpha around 0 5.4%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
   9. Step-by-step derivation

      1. *-commutative 5.4%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

   10. Simplified 5.4%

       \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

   11. Taylor expanded in alpha around 0 7.0%

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

       1. +-commutative 7.0%

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

   13. Simplified 7.0%

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

   14. Taylor expanded in beta around inf 7.0%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 96.4%

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

5. Taylor expanded in beta around 0 71.8%

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

   1. +-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in alpha around 0 44.2%

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

9. Final simplification 44.2%

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

Alternative 17: 47.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in beta around 0 72.1%

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

4. Taylor expanded in alpha around 0 44.5%

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

   1. +-commutative 44.5%

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

6. Simplified 44.5%

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

7. Final simplification 44.5%

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

Alternative 18: 45.5% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Simplified 96.4%

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

5. Taylor expanded in beta around 0 71.8%

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

   1. +-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in alpha around 0 43.5%

   \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + -0.1388888888888889 \cdot \alpha\right)} \]
9. Step-by-step derivation

   1. *-commutative 43.5%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \color{blue}{\alpha \cdot -0.1388888888888889}\right) \]

10. Simplified 43.5%

    \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)} \]

11. Taylor expanded in alpha around 0 44.2%

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

    1. +-commutative 44.2%

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

13. Simplified 44.2%

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

14. Taylor expanded in beta around 0 43.2%

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

15. Final simplification 43.2%

    \[\leadsto 0.08333333333333333 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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)))