Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.8%

Time: 17.1s

Alternatives: 18

Speedup: 3.9×

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: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Simplified 95.2%

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

   1. associate-*r/ 95.3%

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

   2. +-commutative 95.3%

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

5. Applied egg-rr 95.3%

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

   1. times-frac 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := \frac{1 + \alpha}{t_0}\\ \mathbf{if}\;\beta \leq 600000000:\\ \;\;\;\;t_1 \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \end{array} \]

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))) (t_1 (/ (+ 1.0 alpha) t_0)))
   (if (<= beta 600000000.0)
     (* t_1 (/ (+ 1.0 beta) (* (+ alpha (+ beta 3.0)) t_0)))
     (* (/ t_1 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 t_1 = (1.0 + alpha) / t_0;
	double tmp;
	if (beta <= 600000000.0) {
		tmp = t_1 * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = (t_1 / 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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    t_1 = (1.0d0 + alpha) / t_0
    if (beta <= 600000000.0d0) then
        tmp = t_1 * ((1.0d0 + beta) / ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = (t_1 / 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 t_1 = (1.0 + alpha) / t_0;
	double tmp;
	if (beta <= 600000000.0) {
		tmp = t_1 * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = (t_1 / 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)
	t_1 = (1.0 + alpha) / t_0
	tmp = 0
	if beta <= 600000000.0:
		tmp = t_1 * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = (t_1 / t_0) * (1.0 - ((alpha + 2.0) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	t_1 = Float64(Float64(1.0 + alpha) / t_0)
	tmp = 0.0
	if (beta <= 600000000.0)
		tmp = Float64(t_1 * Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(t_1 / 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);
	t_1 = (1.0 + alpha) / t_0;
	tmp = 0.0;
	if (beta <= 600000000.0)
		tmp = t_1 * ((1.0 + beta) / ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = (t_1 / 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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 600000000.0], N[(t$95$1 * N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / 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 < 6e8

   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 99.2%

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

   if 6e8 < beta

   1. Initial program 82.0%

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

   3. Simplified 87.7%

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

      1. associate-*r/ 87.8%

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

      2. +-commutative 87.8%

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

   5. Applied egg-rr 87.8%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around inf 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

   10. Simplified 88.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 600000000:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 100000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{t_0} \cdot \left(1 - \frac{\alpha + 2}{\beta}\right)\\ \end{array} \end{array} \]

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 100000000.0)
     (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ 2.0 beta) (+ beta (+ alpha 2.0)))))
     (* (/ (/ (+ 1.0 alpha) 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 <= 100000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))));
	} else {
		tmp = (((1.0 + alpha) / 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 <= 100000000.0d0) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((2.0d0 + beta) * (beta + (alpha + 2.0d0))))
    else
        tmp = (((1.0d0 + alpha) / 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 <= 100000000.0) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))));
	} else {
		tmp = (((1.0 + alpha) / 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 <= 100000000.0:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))))
	else:
		tmp = (((1.0 + alpha) / 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(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 100000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(2.0 + beta) * Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / 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 <= 100000000.0)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))));
	else
		tmp = (((1.0 + alpha) / 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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 100000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $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 < 1e8

   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 92.0%

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

   4. Taylor expanded in alpha around 0 80.3%

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

   5. Taylor expanded in alpha around 0 65.2%

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

      1. add-log-exp 80.1%

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

      2. associate-*r* 80.1%

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

      3. +-commutative 80.1%

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

      4. exp-prod 80.1%

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

      5. +-commutative 80.1%

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

      6. associate-+r+ 80.1%

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

      7. +-commutative 80.1%

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

      8. +-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. log-pow 80.1%

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

      2. +-commutative 80.1%

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

      3. rem-log-exp 65.2%

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

      4. +-commutative 65.2%

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

      5. *-commutative 65.2%

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

      6. +-commutative 65.2%

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

      7. +-commutative 65.2%

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

      8. +-commutative 65.2%

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

   9. Simplified 65.2%

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

   if 1e8 < beta

   1. Initial program 82.0%

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

   3. Simplified 87.7%

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

      1. associate-*r/ 87.8%

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

      2. +-commutative 87.8%

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

   5. Applied egg-rr 87.8%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around inf 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

   10. Simplified 88.0%

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

4. Final simplification 73.1%

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

Alternative 4: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.5e+23)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ 2.0 beta) (+ beta (+ alpha 2.0)))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5e+23) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.5d+23) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((2.0d0 + beta) * (beta + (alpha + 2.0d0))))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5e+23) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.5e+23:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5e+23)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(2.0 + beta) * Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.5e+23)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((2.0 + beta) * (beta + (alpha + 2.0))));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.5e+23], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.50000000000000004e23

   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 92.2%

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

   4. Taylor expanded in alpha around 0 80.7%

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

   5. Taylor expanded in alpha around 0 66.0%

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

      1. add-log-exp 78.3%

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

      2. associate-*r* 78.3%

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

      3. +-commutative 78.3%

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

      4. exp-prod 78.3%

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

      5. +-commutative 78.3%

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

      6. associate-+r+ 78.3%

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

      7. +-commutative 78.3%

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

      8. +-commutative 78.3%

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

   7. Applied egg-rr 78.3%

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

      1. log-pow 78.3%

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

      2. +-commutative 78.3%

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

      3. rem-log-exp 66.0%

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

      4. +-commutative 66.0%

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

      5. *-commutative 66.0%

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

      6. +-commutative 66.0%

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

      7. +-commutative 66.0%

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

      8. +-commutative 66.0%

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

   9. Simplified 66.0%

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

   if 5.50000000000000004e23 < beta

   1. Initial program 81.2%

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

      1. div-inv 81.2%

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

      2. +-commutative 81.2%

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

      3. *-commutative 81.2%

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

      4. associate-+r+ 81.2%

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

      5. +-commutative 81.2%

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

      6. associate-+r+ 81.2%

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

      7. +-commutative 81.2%

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

      8. +-commutative 81.2%

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

      9. *-commutative 81.2%

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

      10. fma-def 81.2%

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

      11. metadata-eval 81.2%

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

      12. associate-+r+ 81.2%

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

      13. metadata-eval 81.2%

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

      14. associate-+r+ 81.2%

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

   3. Applied egg-rr 81.2%

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

      1. associate-*l/ 81.2%

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

      2. associate-*r/ 81.2%

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

      3. *-rgt-identity 81.2%

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

      4. +-commutative 81.2%

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

      5. fma-udef 81.2%

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

      6. distribute-lft1-in 81.2%

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

      7. +-commutative 81.2%

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

      8. *-commutative 81.2%

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

      9. distribute-rgt1-in 81.2%

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

      10. +-commutative 81.2%

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

      11. +-commutative 81.2%

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

      12. associate-*l/ 99.7%

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

      13. +-commutative 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 88.1%

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

   7. Taylor expanded in alpha around 0 88.1%

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

      1. associate-+r+ 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+r+ 88.1%

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

   9. Simplified 88.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 5: 97.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.5)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (+ 1.0 alpha) (+ beta (+ alpha 2.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / (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 <= 5.5d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((1.0d0 + alpha) / (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 <= 5.5) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.5:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + 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 <= 5.5)
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((1.0 + alpha) / (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, 5.5], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5

   1. Initial program 99.9%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 97.5%

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

      1. +-commutative 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in alpha around 0 63.4%

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

      1. *-commutative 63.4%

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

   9. Simplified 63.4%

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

   if 5.5 < 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 87.8%

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

   4. Taylor expanded in beta around inf 86.2%

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

      1. un-div-inv 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. associate-+r+ 86.3%

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

   6. Applied egg-rr 86.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \]

Alternative 6: 97.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.9:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.9)
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))
    (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.9) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.9d0) then
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.9) {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.9:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.9)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + 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 = ((1.0 + alpha) / (alpha + (2.0 + beta))) * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.9], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9000000000000004

   1. Initial program 99.9%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in beta around 0 97.5%

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

      1. +-commutative 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in alpha around 0 63.4%

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

      1. *-commutative 63.4%

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

   9. Simplified 63.4%

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

   if 4.9000000000000004 < beta

   1. Initial program 82.2%

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

      1. div-inv 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. +-commutative 82.2%

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

      6. associate-+r+ 82.2%

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

      7. +-commutative 82.2%

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

      8. +-commutative 82.2%

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

      9. *-commutative 82.2%

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

      10. fma-def 82.2%

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

      11. metadata-eval 82.2%

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

      12. associate-+r+ 82.2%

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

      13. metadata-eval 82.2%

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

      14. associate-+r+ 82.2%

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

   3. Applied egg-rr 82.2%

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

      1. associate-*l/ 82.1%

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

      2. associate-*r/ 82.2%

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

      3. *-rgt-identity 82.2%

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

      4. +-commutative 82.2%

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

      5. fma-udef 82.2%

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

      6. distribute-lft1-in 82.2%

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

      7. +-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 82.2%

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

      10. +-commutative 82.2%

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

      11. +-commutative 82.2%

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

      12. associate-*l/ 99.7%

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

      13. +-commutative 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 86.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.9:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 7: 97.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}\\ \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;t_0 \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))))
   (if (<= beta 2.8)
     (* t_0 (+ 0.16666666666666666 (* alpha -0.1388888888888889)))
     (/ t_0 (+ alpha (+ beta 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (1.0 + alpha) / (alpha + (2.0 + beta));
	double tmp;
	if (beta <= 2.8) {
		tmp = t_0 * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = t_0 / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + alpha) / (alpha + (2.0d0 + beta))
    if (beta <= 2.8d0) then
        tmp = t_0 * (0.16666666666666666d0 + (alpha * (-0.1388888888888889d0)))
    else
        tmp = t_0 / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (1.0 + alpha) / (alpha + (2.0 + beta));
	double tmp;
	if (beta <= 2.8) {
		tmp = t_0 * (0.16666666666666666 + (alpha * -0.1388888888888889));
	} else {
		tmp = t_0 / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (1.0 + alpha) / (alpha + (2.0 + beta))
	tmp = 0
	if beta <= 2.8:
		tmp = t_0 * (0.16666666666666666 + (alpha * -0.1388888888888889))
	else:
		tmp = t_0 / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta)))
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(t_0 * Float64(0.16666666666666666 + Float64(alpha * -0.1388888888888889)));
	else
		tmp = Float64(t_0 / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + alpha) / (alpha + (2.0 + beta));
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = t_0 * (0.16666666666666666 + (alpha * -0.1388888888888889));
	else
		tmp = t_0 / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.8], N[(t$95$0 * N[(0.16666666666666666 + N[(alpha * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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 99.2%

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

   4. Taylor expanded in beta around 0 97.5%

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

      1. +-commutative 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in alpha around 0 63.4%

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

      1. *-commutative 63.4%

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

   9. Simplified 63.4%

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

   if 2.7999999999999998 < beta

   1. Initial program 82.2%

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

      1. div-inv 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. +-commutative 82.2%

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

      6. associate-+r+ 82.2%

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

      7. +-commutative 82.2%

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

      8. +-commutative 82.2%

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

      9. *-commutative 82.2%

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

      10. fma-def 82.2%

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

      11. metadata-eval 82.2%

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

      12. associate-+r+ 82.2%

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

      13. metadata-eval 82.2%

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

      14. associate-+r+ 82.2%

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

   3. Applied egg-rr 82.2%

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

      1. associate-*l/ 82.1%

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

      2. associate-*r/ 82.2%

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

      3. *-rgt-identity 82.2%

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

      4. +-commutative 82.2%

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

      5. fma-udef 82.2%

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

      6. distribute-lft1-in 82.2%

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

      7. +-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 82.2%

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

      10. +-commutative 82.2%

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

      11. +-commutative 82.2%

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

      12. associate-*l/ 99.7%

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

      13. +-commutative 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 86.9%

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

   7. Taylor expanded in alpha around 0 86.9%

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

      1. associate-+r+ 86.9%

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

      2. +-commutative 86.9%

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

      3. associate-+r+ 86.9%

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

   9. Simplified 86.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8: 98.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2e+23)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ 6.0 (* beta (+ beta 5.0))))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta))) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+23) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d+23) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (6.0d0 + (beta * (beta + 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (2.0d0 + beta))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+23) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2e+23:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2e+23)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(6.0 + Float64(beta * Float64(beta + 5.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2e+23)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (6.0 + (beta * (beta + 5.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (2.0 + beta))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2e+23], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(beta * N[(beta + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.19999999999999983e23

   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 99.2%

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

   4. Taylor expanded in beta around 0 99.3%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\color{blue}{\beta \cdot \left(5 + 2 \cdot \alpha\right) + \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right) + {\beta}^{2}\right)}} \]

   5. Taylor expanded in alpha around 0 65.1%

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

      1. associate-/r* 65.1%

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

      2. +-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. distribute-rgt-out 65.1%

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

   7. Simplified 65.1%

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

   if 5.19999999999999983e23 < beta

   1. Initial program 81.2%

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

      1. div-inv 81.2%

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

      2. +-commutative 81.2%

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

      3. *-commutative 81.2%

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

      4. associate-+r+ 81.2%

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

      5. +-commutative 81.2%

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

      6. associate-+r+ 81.2%

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

      7. +-commutative 81.2%

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

      8. +-commutative 81.2%

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

      9. *-commutative 81.2%

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

      10. fma-def 81.2%

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

      11. metadata-eval 81.2%

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

      12. associate-+r+ 81.2%

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

      13. metadata-eval 81.2%

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

      14. associate-+r+ 81.2%

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

   3. Applied egg-rr 81.2%

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

      1. associate-*l/ 81.2%

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

      2. associate-*r/ 81.2%

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

      3. *-rgt-identity 81.2%

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

      4. +-commutative 81.2%

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

      5. fma-udef 81.2%

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

      6. distribute-lft1-in 81.2%

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

      7. +-commutative 81.2%

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

      8. *-commutative 81.2%

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

      9. distribute-rgt1-in 81.2%

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

      10. +-commutative 81.2%

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

      11. +-commutative 81.2%

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

      12. associate-*l/ 99.7%

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

      13. +-commutative 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 88.1%

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

   7. Taylor expanded in alpha around 0 88.1%

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

      1. associate-+r+ 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+r+ 88.1%

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

   9. Simplified 88.1%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{6 + \beta \cdot \left(\beta + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9: 97.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 6.0 (* beta 5.0))))
   (/ (/ (+ 1.0 alpha) (+ beta (+ alpha 2.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (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 <= 5.2d0) then
        tmp = (1.0d0 + beta) / ((2.0d0 + beta) * (6.0d0 + (beta * 5.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (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 <= 5.2) {
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * 5.0)));
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * 5.0)))
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(6.0 + Float64(beta * 5.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + 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 <= 5.2)
		tmp = (1.0 + beta) / ((2.0 + beta) * (6.0 + (beta * 5.0)));
	else
		tmp = ((1.0 + alpha) / (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, 5.2], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(6.0 + N[(beta * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   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 92.0%

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

   4. Taylor expanded in beta around 0 91.3%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \color{blue}{\left(\beta \cdot \left(5 + 2 \cdot \alpha\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   5. Taylor expanded in alpha around 0 63.9%

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

      1. *-commutative 63.9%

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

   7. Simplified 63.9%

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

   if 5.20000000000000018 < 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 87.8%

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

   4. Taylor expanded in beta around inf 86.2%

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

      1. un-div-inv 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. associate-+r+ 86.3%

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

   6. Applied egg-rr 86.3%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \]

Alternative 10: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ (/ (+ 1.0 alpha) (+ beta (+ alpha 2.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (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 <= 3.2d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / (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 <= 3.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	else:
		tmp = ((1.0 + alpha) / (beta + (alpha + 2.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(beta + 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 <= 3.2)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	else
		tmp = ((1.0 + alpha) / (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, 3.2], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2000000000000002

   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 92.0%

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

   4. Taylor expanded in alpha around 0 80.7%

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

   5. Taylor expanded in alpha around 0 65.6%

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

   6. Taylor expanded in beta around 0 64.3%

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

      1. +-commutative 64.3%

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

   8. Simplified 64.3%

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

   if 3.2000000000000002 < 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 87.8%

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

   4. Taylor expanded in beta around inf 86.2%

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

      1. un-div-inv 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. associate-+r+ 86.3%

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

   6. Applied egg-rr 86.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \]

Alternative 11: 91.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ 1.0 (* beta (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 / (beta * (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = 1.0 / (beta * (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	else:
		tmp = 1.0 / (beta * (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	else
		tmp = 1.0 / (beta * (2.0 + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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 92.0%

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

   4. Taylor expanded in alpha around 0 80.7%

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

   5. Taylor expanded in alpha around 0 65.6%

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

   6. Taylor expanded in beta around 0 64.3%

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

      1. +-commutative 64.3%

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

   8. Simplified 64.3%

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

   if 2.7999999999999998 < 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 87.8%

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

   4. Taylor expanded in beta around inf 86.2%

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

   5. Taylor expanded in alpha around 0 77.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \end{array} \]

Alternative 12: 91.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.45)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.45d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.45:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.45)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.45)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.45], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   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 92.0%

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

   4. Taylor expanded in alpha around 0 80.7%

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

   5. Taylor expanded in alpha around 0 65.6%

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

   6. Taylor expanded in beta around 0 64.3%

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

      1. +-commutative 64.3%

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

   8. Simplified 64.3%

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

   if 2.4500000000000002 < beta

   1. Initial program 82.2%

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

      1. div-inv 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. +-commutative 82.2%

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

      6. associate-+r+ 82.2%

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

      7. +-commutative 82.2%

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

      8. +-commutative 82.2%

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

      9. *-commutative 82.2%

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

      10. fma-def 82.2%

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

      11. metadata-eval 82.2%

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

      12. associate-+r+ 82.2%

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

      13. metadata-eval 82.2%

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

      14. associate-+r+ 82.2%

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

   3. Applied egg-rr 82.2%

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

      1. associate-*l/ 82.1%

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

      2. associate-*r/ 82.2%

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

      3. *-rgt-identity 82.2%

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

      4. +-commutative 82.2%

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

      5. fma-udef 82.2%

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

      6. distribute-lft1-in 82.2%

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

      7. +-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 82.2%

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

      10. +-commutative 82.2%

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

      11. +-commutative 82.2%

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

      12. associate-*l/ 99.7%

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

      13. +-commutative 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 86.3%

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

   7. Taylor expanded in alpha around 0 77.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 13: 91.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in alpha around 0 80.7%

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

   5. Taylor expanded in alpha around 0 65.6%

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

   6. Taylor expanded in beta around 0 64.3%

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

      1. +-commutative 64.3%

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

   8. Simplified 64.3%

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

   if 2.60000000000000009 < 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 87.8%

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

   4. Taylor expanded in beta around inf 86.2%

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

      1. un-div-inv 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. associate-+r+ 86.3%

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

   6. Applied egg-rr 86.3%

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

   7. Taylor expanded in alpha around 0 77.7%

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

4. Final simplification 69.0%

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

Alternative 14: 96.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ 0.16666666666666666 (+ alpha 2.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.2d0) then
        tmp = 0.16666666666666666d0 / (alpha + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 0.16666666666666666 / (alpha + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = 0.16666666666666666 / (alpha + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(0.16666666666666666 / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = 0.16666666666666666 / (alpha + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.2], N[(0.16666666666666666 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   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 92.0%

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

   4. Taylor expanded in alpha around 0 80.7%

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

   5. Taylor expanded in alpha around 0 65.6%

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

   6. Taylor expanded in beta around 0 64.3%

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

      1. +-commutative 64.3%

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

   8. Simplified 64.3%

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

   if 4.20000000000000018 < beta

   1. Initial program 82.2%

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

      1. div-inv 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-+r+ 82.2%

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

      5. +-commutative 82.2%

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

      6. associate-+r+ 82.2%

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

      7. +-commutative 82.2%

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

      8. +-commutative 82.2%

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

      9. *-commutative 82.2%

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

      10. fma-def 82.2%

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

      11. metadata-eval 82.2%

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

      12. associate-+r+ 82.2%

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

      13. metadata-eval 82.2%

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

      14. associate-+r+ 82.2%

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

   3. Applied egg-rr 82.2%

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

      1. associate-*l/ 82.1%

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

      2. associate-*r/ 82.2%

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

      3. *-rgt-identity 82.2%

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

      4. +-commutative 82.2%

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

      5. fma-udef 82.2%

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

      6. distribute-lft1-in 82.2%

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

      7. +-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 82.2%

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

      10. +-commutative 82.2%

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

      11. +-commutative 82.2%

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

      12. associate-*l/ 99.7%

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

      13. +-commutative 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 86.3%

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

   7. Taylor expanded in beta around inf 86.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 15: 46.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.16666666666666666 \cdot \frac{1}{2 + \beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 * (1.0 / (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 * (1.0d0 / (2.0d0 + beta))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 * (1.0 / (2.0 + beta));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 * (1.0 / (2.0 + beta))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 * Float64(1.0 / Float64(2.0 + beta)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 * (1.0 / (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[(1.0 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Simplified 95.2%

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

4. Taylor expanded in beta around 0 70.5%

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

   1. +-commutative 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in alpha around 0 43.1%

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

8. Taylor expanded in alpha around 0 43.5%

   \[\leadsto \color{blue}{\frac{1}{2 + \beta}} \cdot 0.16666666666666666 \]

9. Final simplification 43.5%

   \[\leadsto 0.16666666666666666 \cdot \frac{1}{2 + \beta} \]

Alternative 16: 46.6% accurate, 6.9× 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 99.2%

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

   4. Taylor expanded in beta around 0 97.5%

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

      1. +-commutative 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in alpha around 0 63.4%

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

   8. Taylor expanded in beta around 0 63.3%

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

   if 2 < 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 87.8%

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

   4. Taylor expanded in beta around 0 20.7%

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

      1. +-commutative 20.7%

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

   6. Simplified 20.7%

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

   7. Taylor expanded in alpha around 0 6.9%

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

   8. Taylor expanded in beta around inf 6.9%

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

4. Final simplification 43.5%

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

Alternative 17: 46.7% 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 93.7%

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

3. Simplified 95.2%

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

4. Taylor expanded in beta around 0 70.5%

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

   1. +-commutative 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in alpha around 0 43.5%

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

8. Final simplification 43.5%

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

Alternative 18: 44.9% 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 93.7%

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

3. Simplified 95.2%

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

4. Taylor expanded in beta around 0 70.5%

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

   1. +-commutative 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in alpha around 0 43.5%

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

8. Taylor expanded in beta around 0 42.4%

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

9. Final simplification 42.4%

   \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023332 
(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)))