Octave 3.8, jcobi/3

Percentage Accurate: 94.7% → 99.8%

Time: 18.3s

Alternatives: 21

Speedup: 2.9×

• could not determine a ground truth

  1. alpha = 3.989083864485918e-271
  2. beta = 7.67414387512699e+35

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: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Simplified 83.2%

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

   1. times-frac 96.4%

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

   2. +-commutative 96.4%

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

6. Applied egg-rr 96.4%

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

   1. associate-*r/ 96.4%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-+r+ 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+r+ 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5.6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 5.6)
     (/ (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 3.0))) t_0)
     (/ (/ (* (+ 1.0 alpha) (+ 1.0 (/ (- -2.0 alpha) beta))) t_0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5.6) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) * (1.0 + ((-2.0 - alpha) / beta))) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 5.6d0) then
        tmp = ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 3.0d0))) / t_0
    else
        tmp = (((1.0d0 + alpha) * (1.0d0 + (((-2.0d0) - alpha) / beta))) / t_0) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5.6) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) * (1.0 + ((-2.0 - alpha) / beta))) / t_0) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5.6:
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0
	else:
		tmp = (((1.0 + alpha) * (1.0 + ((-2.0 - alpha) / beta))) / t_0) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5.6)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + Float64(Float64(-2.0 - alpha) / beta))) / t_0) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5.6)
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	else
		tmp = (((1.0 + alpha) * (1.0 + ((-2.0 - alpha) / beta))) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.6], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5999999999999996

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around 0 98.0%

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

   if 5.5999999999999996 < beta

   1. Initial program 84.3%

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

   3. Simplified 67.0%

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

      1. times-frac 90.7%

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

      2. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. associate-*r/ 90.8%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.8%

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

       2. associate-+r+ 99.8%

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

       3. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in beta around inf 87.3%

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

       1. associate-*r/ 87.3%

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

       2. distribute-lft-in 87.3%

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

       3. metadata-eval 87.3%

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

       4. mul-1-neg 87.3%

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

       5. unsub-neg 87.3%

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

   15. Simplified 87.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 11.5:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 11.5)
     (/ (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 3.0))) t_0)
     (* (/ (/ (+ 1.0 alpha) t_0) t_0) (+ 1.0 (/ (- -2.0 alpha) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 11.5) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (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) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 11.5d0) then
        tmp = ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 3.0d0))) / t_0
    else
        tmp = (((1.0d0 + alpha) / t_0) / t_0) * (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 t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 11.5) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 11.5:
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0
	else:
		tmp = (((1.0 + alpha) / t_0) / t_0) * (1.0 + ((-2.0 - alpha) / beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 11.5)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0) * Float64(1.0 + Float64(Float64(-2.0 - alpha) / beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 11.5)
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	else
		tmp = (((1.0 + alpha) / t_0) / t_0) * (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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 11.5], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 11.5

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around 0 98.0%

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

   if 11.5 < beta

   1. Initial program 84.3%

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

   3. Simplified 67.0%

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

      1. times-frac 90.7%

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

      2. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. associate-*r/ 90.8%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 87.4%

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

       1. associate-*r/ 87.4%

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

       2. distribute-lft-in 87.4%

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

       3. metadata-eval 87.4%

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

       4. mul-1-neg 87.4%

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

   11. Simplified 87.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 11.5:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 9.2)
     (/ (/ (+ 1.0 alpha) (* (+ alpha 2.0) (+ alpha 3.0))) t_0)
     (/ (* (/ (+ 1.0 alpha) t_0) (+ 1.0 (/ (- -2.0 alpha) beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9.2) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 9.2d0) then
        tmp = ((1.0d0 + alpha) / ((alpha + 2.0d0) * (alpha + 3.0d0))) / t_0
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 + (((-2.0d0) - alpha) / beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9.2) {
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 9.2:
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 9.2)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 + Float64(Float64(-2.0 - alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 9.2)
		tmp = ((1.0 + alpha) / ((alpha + 2.0) * (alpha + 3.0))) / t_0;
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-2.0 - alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.2], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-2.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.1999999999999993

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around 0 98.0%

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

   if 9.1999999999999993 < beta

   1. Initial program 84.3%

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

   3. Simplified 67.0%

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

      1. times-frac 90.7%

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

      2. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. associate-*r/ 90.8%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in beta around inf 87.4%

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

       1. associate-*r/ 87.4%

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

       2. distribute-lft-in 87.4%

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

       3. metadata-eval 87.4%

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

       4. mul-1-neg 87.4%

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

   13. Simplified 87.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.25e+15)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/
      (* (/ (+ 1.0 beta) (+ alpha (+ beta 3.0))) (/ (+ 1.0 alpha) beta))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.25e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + beta) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.25d+15) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = (((1.0d0 + beta) / (alpha + (beta + 3.0d0))) * ((1.0d0 + alpha) / beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.25e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + beta) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.25e+15:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = (((1.0 + beta) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.25e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 + alpha) / beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.25e+15)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = (((1.0 + beta) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.25e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25e15

   1. Initial program 99.8%

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

   3. Simplified 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. Add Preprocessing
   5. Step-by-step derivation

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in alpha around 0 67.6%

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

       1. +-commutative 67.6%

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

       2. +-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 1.25e15 < beta

   1. Initial program 83.6%

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

   3. Simplified 65.5%

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

      1. times-frac 90.3%

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

      2. +-commutative 90.3%

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

   6. Applied egg-rr 90.3%

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

      1. associate-*r/ 90.4%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in beta around inf 88.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Simplified 83.2%

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

   1. times-frac 96.4%

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

   2. +-commutative 96.4%

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

6. Applied egg-rr 96.4%

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

   1. associate-*r/ 96.4%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-+r+ 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 7: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.5e+15)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/ (/ (+ 1.0 alpha) t_0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.5e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.5d+15) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.5e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.5e+15:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = ((1.0 + alpha) / t_0) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.5e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.5e+15)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = ((1.0 + alpha) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.5e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e15

   1. Initial program 99.8%

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

   3. Simplified 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. Add Preprocessing
   5. Step-by-step derivation

      1. times-frac 99.5%

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

      2. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in alpha around 0 67.6%

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

       1. +-commutative 67.6%

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

       2. +-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 1.5e15 < beta

   1. Initial program 83.6%

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

   3. Simplified 65.5%

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

      1. times-frac 90.3%

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

      2. +-commutative 90.3%

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

   6. Applied egg-rr 90.3%

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

      1. associate-*r/ 90.4%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. associate-+r+ 99.8%

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

       3. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in beta around inf 88.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 4.0)
     (/ 1.0 (* t_0 (- 6.0 alpha)))
     (/ (/ (+ 1.0 alpha) t_0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.0) {
		tmp = 1.0 / (t_0 * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 4.0d0) then
        tmp = 1.0d0 / (t_0 * (6.0d0 - alpha))
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.0) {
		tmp = 1.0 / (t_0 * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 4.0:
		tmp = 1.0 / (t_0 * (6.0 - alpha))
	else:
		tmp = ((1.0 + alpha) / t_0) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(1.0 / Float64(t_0 * Float64(6.0 - alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 1.0 / (t_0 * (6.0 - alpha));
	else
		tmp = ((1.0 + alpha) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.0], N[(1.0 / N[(t$95$0 * N[(6.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 83.2%

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

       1. mul-1-neg 83.2%

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

   12. Simplified 83.2%

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

   if 4 < beta

   1. Initial program 84.3%

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

   3. Simplified 67.0%

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

      1. times-frac 90.7%

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

      2. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. associate-*r/ 90.8%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.8%

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

       2. associate-+r+ 99.8%

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

       3. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in beta around inf 86.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1.3e+16)
     (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
     (/ (/ (+ 1.0 alpha) t_0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.3e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 1.3d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / t_0) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1.3e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1.3e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / t_0) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1.3e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1.3e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.3e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3e16

   1. Initial program 99.8%

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

      1. associate-/l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-+l+ 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-+l+ 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 85.6%

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

      1. +-commutative 85.6%

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

   7. Simplified 85.6%

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

   8. Taylor expanded in alpha around 0 66.3%

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

      1. +-commutative 66.3%

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

   10. Simplified 66.3%

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

   if 1.3e16 < beta

   1. Initial program 83.6%

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

   3. Simplified 65.5%

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

      1. times-frac 90.3%

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

      2. +-commutative 90.3%

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

   6. Applied egg-rr 90.3%

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

      1. associate-*r/ 90.4%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. associate-+r+ 99.8%

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

       3. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in beta around inf 88.3%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) 6.0))
   (/ (* (+ 1.0 alpha) (/ 1.0 beta)) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) * (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 <= 6.2d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * 6.0d0)
    else
        tmp = ((1.0d0 + alpha) * (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 <= 6.2) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0)
	else:
		tmp = ((1.0 + alpha) * (1.0 / beta)) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * 6.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / beta)) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	else
		tmp = ((1.0 + alpha) * (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, 6.2], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 66.6%

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

   if 6.20000000000000018 < beta

   1. Initial program 84.3%

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

   3. Taylor expanded in beta around inf 85.9%

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

   4. Taylor expanded in alpha around 0 85.8%

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

      1. +-commutative 85.8%

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

   6. Simplified 85.8%

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

      1. div-inv 85.8%

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

   8. Applied egg-rr 85.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.5)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) 6.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.5d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * 6.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.5:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * 6.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.5)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.5], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 66.6%

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

   if 5.5 < beta

   1. Initial program 84.3%

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

   3. Taylor expanded in beta around inf 85.9%

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

      1. metadata-eval 85.9%

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

      2. associate-+l+ 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

      5. associate-+r+ 85.9%

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

      6. *-un-lft-identity 85.9%

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

      7. fma-define 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. fma-undefine 85.9%

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

      2. *-lft-identity 85.9%

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

      3. +-commutative 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.9:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.9)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) (- 6.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.9) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.9d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * (6.0d0 - alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.9) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.9:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.9)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(6.0 - alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.9)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha));
	else
		tmp = ((1.0 + alpha) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.9], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(6.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9000000000000004

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 83.2%

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

       1. mul-1-neg 83.2%

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

   12. Simplified 83.2%

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

   if 4.9000000000000004 < beta

   1. Initial program 84.3%

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

   3. Taylor expanded in beta around inf 85.9%

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

      1. metadata-eval 85.9%

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

      2. associate-+l+ 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

      5. associate-+r+ 85.9%

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

      6. *-un-lft-identity 85.9%

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

      7. fma-define 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. fma-undefine 85.9%

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

      2. *-lft-identity 85.9%

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

      3. +-commutative 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.9:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) 6.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} 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 <= 8.2d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * 6.0d0)
    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 <= 8.2) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.2:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * 6.0));
	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 <= 8.2)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	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, 8.2], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.1999999999999993

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 66.6%

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

   if 8.1999999999999993 < beta

   1. Initial program 84.3%

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

   3. Taylor expanded in beta around inf 85.9%

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

   4. Taylor expanded in alpha around -inf 80.1%

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

      1. associate-*r* 80.1%

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

      2. mul-1-neg 80.1%

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

      3. sub-neg 80.1%

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

      4. associate-*r/ 80.1%

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

      5. mul-1-neg 80.1%

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

      6. distribute-neg-in 80.1%

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

      7. unsub-neg 80.1%

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

      8. metadata-eval 80.1%

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

      9. metadata-eval 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in beta around inf 85.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.6:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.6)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) 6.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.6) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.6d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * 6.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.6) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.6:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.6)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * 6.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.6)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.6], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5999999999999996

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 66.6%

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

   if 5.5999999999999996 < beta

   1. Initial program 84.3%

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

   3. Taylor expanded in beta around inf 85.9%

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

   4. Taylor expanded in alpha around 0 85.8%

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

      1. +-commutative 85.8%

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

   6. Simplified 85.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.6:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4) (/ 1.0 (* (+ beta 2.0) 6.0)) (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} 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 <= 5.4d0) then
        tmp = 1.0d0 / ((beta + 2.0d0) * 6.0d0)
    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 <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 1.0 / ((beta + 2.0) * 6.0)
	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 <= 5.4)
		tmp = Float64(1.0 / Float64(Float64(beta + 2.0) * 6.0));
	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 <= 5.4)
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	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, 5.4], N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 64.9%

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

   if 5.4000000000000004 < beta

   1. Initial program 84.3%

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

   3. Taylor expanded in beta around inf 85.9%

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

   4. Taylor expanded in alpha around 0 80.4%

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

      1. +-commutative 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4) (/ 1.0 (* (+ beta 2.0) 6.0)) (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} 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 <= 5.4d0) then
        tmp = 1.0d0 / ((beta + 2.0d0) * 6.0d0)
    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 <= 5.4) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.4:
		tmp = 1.0 / ((beta + 2.0) * 6.0)
	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 <= 5.4)
		tmp = Float64(1.0 / Float64(Float64(beta + 2.0) * 6.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	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, 5.4], N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+l+ 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.8%

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

      9. associate-+r+ 91.8%

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

      10. +-commutative 91.8%

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

      11. associate-/l/ 99.5%

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

      12. clear-num 99.5%

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

      13. inv-pow 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-undefine 99.5%

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

      11. +-commutative 99.5%

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

      12. *-commutative 99.5%

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

      13. +-commutative 99.5%

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

      14. associate-+r+ 99.5%

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

      15. distribute-lft1-in 99.5%

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

      16. +-commutative 99.5%

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

      17. +-commutative 99.5%

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

      18. +-commutative 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in beta around 0 98.0%

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

   10. Taylor expanded in alpha around 0 64.9%

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

   if 5.4000000000000004 < beta

   1. Initial program 84.3%

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

   3. Taylor expanded in beta around inf 85.9%

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

      1. clear-num 85.2%

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

      2. inv-pow 85.2%

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

      3. metadata-eval 85.2%

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

      4. associate-+l+ 85.2%

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

      5. metadata-eval 85.2%

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

      6. +-commutative 85.2%

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

      7. +-commutative 85.2%

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

      8. associate-+r+ 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. unpow-1 85.2%

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

      2. associate-/r/ 85.2%

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

      3. +-commutative 85.2%

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

      4. +-commutative 85.2%

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

   7. Simplified 85.2%

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

   8. Taylor expanded in alpha around 0 80.4%

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

      1. associate-/r* 81.0%

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

      2. +-commutative 81.0%

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

   10. Simplified 81.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 96.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 1.0 (* (+ beta 2.0) 6.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} 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 <= 8.0d0) then
        tmp = 1.0d0 / ((beta + 2.0d0) * 6.0d0)
    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 <= 8.0) {
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 1.0 / ((beta + 2.0) * 6.0)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(1.0 / Float64(Float64(beta + 2.0) * 6.0));
	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 <= 8.0)
		tmp = 1.0 / ((beta + 2.0) * 6.0);
	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, 8.0], N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8

   1. Initial program 99.8%

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

   3. Simplified 91.8%

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

      1. associate-+r+ 91.8%

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

      2. fma-undefine 91.8%

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

      3. *-commutative 91.8%

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

      4. associate-+l+ 91.8%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      5. +-commutative 91.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      6. associate-+l+ 91.8%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      7. *-commutative 91.8%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      8. associate-*r* 91.8%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      9. associate-+r+ 91.8%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      10. +-commutative 91.8%

          \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      11. associate-/l/ 99.5%

          \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      12. clear-num 99.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      13. inv-pow 99.5%

          \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 99.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      4. +-commutative 99.5%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      5. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      6. associate-+r+ 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      7. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \color{blue}{\left(\beta + \alpha\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      9. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

      10. fma-undefine 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

      11. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      12. *-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      13. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      14. associate-+r+ 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      15. distribute-lft1-in 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      16. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

      17. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      18. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

      19. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

   9. Taylor expanded in beta around 0 98.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]

   10. Taylor expanded in alpha around 0 64.9%

       \[\leadsto \frac{1}{\color{blue}{6 \cdot \left(2 + \beta\right)}} \]

   if 8 < beta

   1. Initial program 84.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 85.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in alpha around -inf 80.1%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{-1 \cdot \left(\alpha \cdot \left(-1 \cdot \frac{3 + \beta}{\alpha} - 1\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\left(-1 \cdot \alpha\right) \cdot \left(-1 \cdot \frac{3 + \beta}{\alpha} - 1\right)}} \]

      2. mul-1-neg 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\left(-\alpha\right)} \cdot \left(-1 \cdot \frac{3 + \beta}{\alpha} - 1\right)} \]

      3. sub-neg 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\left(-\alpha\right) \cdot \color{blue}{\left(-1 \cdot \frac{3 + \beta}{\alpha} + \left(-1\right)\right)}} \]

      4. associate-*r/ 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\left(-\alpha\right) \cdot \left(\color{blue}{\frac{-1 \cdot \left(3 + \beta\right)}{\alpha}} + \left(-1\right)\right)} \]

      5. mul-1-neg 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\left(-\alpha\right) \cdot \left(\frac{\color{blue}{-\left(3 + \beta\right)}}{\alpha} + \left(-1\right)\right)} \]

      6. distribute-neg-in 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\left(-\alpha\right) \cdot \left(\frac{\color{blue}{\left(-3\right) + \left(-\beta\right)}}{\alpha} + \left(-1\right)\right)} \]

      7. unsub-neg 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\left(-\alpha\right) \cdot \left(\frac{\color{blue}{\left(-3\right) - \beta}}{\alpha} + \left(-1\right)\right)} \]

      8. metadata-eval 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\left(-\alpha\right) \cdot \left(\frac{\color{blue}{-3} - \beta}{\alpha} + \left(-1\right)\right)} \]

      9. metadata-eval 80.1%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\left(-\alpha\right) \cdot \left(\frac{-3 - \beta}{\alpha} + \color{blue}{-1}\right)} \]

   6. Simplified 80.1%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\left(-\alpha\right) \cdot \left(\frac{-3 - \beta}{\alpha} + -1\right)}} \]

   7. Taylor expanded in beta around inf 85.7%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) 0.08333333333333333 (/ 0.16666666666666666 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 0.16666666666666666d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 0.16666666666666666 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.08333333333333333
	else:
		tmp = 0.16666666666666666 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(0.16666666666666666 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.08333333333333333;
	else
		tmp = 0.16666666666666666 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.08333333333333333, N[(0.16666666666666666 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Simplified 91.8%

      \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ 91.8%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      2. fma-undefine 91.8%

         \[\leadsto \frac{\left(\alpha + \beta\right) + \color{blue}{\left(\alpha \cdot \beta + 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      3. *-commutative 91.8%

         \[\leadsto \frac{\left(\alpha + \beta\right) + \left(\color{blue}{\beta \cdot \alpha} + 1\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      4. associate-+l+ 91.8%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      5. +-commutative 91.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      6. associate-+l+ 91.8%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      7. *-commutative 91.8%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      8. associate-*r* 91.8%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      9. associate-+r+ 91.8%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      10. +-commutative 91.8%

          \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      11. associate-/l/ 99.5%

          \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      12. clear-num 99.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      13. inv-pow 99.5%

          \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 99.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      2. associate-/l* 99.5%

         \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      4. +-commutative 99.5%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      5. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      6. associate-+r+ 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      7. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \color{blue}{\left(\beta + \alpha\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      9. +-commutative 99.5%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

      10. fma-undefine 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

      11. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      12. *-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      13. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      14. associate-+r+ 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      15. distribute-lft1-in 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      16. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

      17. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      18. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

      19. +-commutative 99.5%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

   9. Taylor expanded in beta around 0 98.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]

   10. Taylor expanded in alpha around 0 64.9%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
   11. Step-by-step derivation

       1. +-commutative 64.9%

          \[\leadsto \frac{0.16666666666666666}{\color{blue}{\beta + 2}} \]

   12. Simplified 64.9%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

   13. Taylor expanded in beta around 0 64.9%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2 < beta

   1. Initial program 84.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ 67.0%

         \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      2. fma-undefine 67.0%

         \[\leadsto \frac{\left(\alpha + \beta\right) + \color{blue}{\left(\alpha \cdot \beta + 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      3. *-commutative 67.0%

         \[\leadsto \frac{\left(\alpha + \beta\right) + \left(\color{blue}{\beta \cdot \alpha} + 1\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      4. associate-+l+ 67.0%

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      5. +-commutative 67.0%

         \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      6. associate-+l+ 67.0%

         \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      7. *-commutative 67.0%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

      8. associate-*r* 67.0%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      9. associate-+r+ 67.0%

         \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      10. +-commutative 67.0%

          \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      11. associate-/l/ 81.3%

          \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      12. clear-num 81.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      13. inv-pow 81.3%

          \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

   6. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 81.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      2. associate-/l* 83.6%

         \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

      3. +-commutative 83.6%

         \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      4. +-commutative 83.6%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      5. +-commutative 83.6%

         \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      6. associate-+r+ 83.6%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      7. +-commutative 83.6%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      8. +-commutative 83.6%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \color{blue}{\left(\beta + \alpha\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

      9. +-commutative 83.6%

         \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

      10. fma-undefine 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

      11. +-commutative 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      12. *-commutative 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      13. +-commutative 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

      14. associate-+r+ 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      15. distribute-lft1-in 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      16. +-commutative 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

      17. +-commutative 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

      18. +-commutative 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

      19. +-commutative 83.6%

          \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

   8. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

   9. Taylor expanded in beta around 0 16.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]

   10. Taylor expanded in alpha around 0 7.5%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
   11. Step-by-step derivation

       1. +-commutative 7.5%

          \[\leadsto \frac{0.16666666666666666}{\color{blue}{\beta + 2}} \]

   12. Simplified 7.5%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

   13. Taylor expanded in beta around inf 7.5%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 47.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\left(\beta + 2\right) \cdot 6} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 1.0 (* (+ beta 2.0) 6.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / ((beta + 2.0) * 6.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0 / ((beta + 2.0d0) * 6.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 1.0 / ((beta + 2.0) * 6.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 1.0 / ((beta + 2.0) * 6.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(1.0 / Float64(Float64(beta + 2.0) * 6.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 1.0 / ((beta + 2.0) * 6.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Simplified 83.2%

   \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   2. fma-undefine 83.2%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \color{blue}{\left(\alpha \cdot \beta + 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   3. *-commutative 83.2%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \left(\color{blue}{\beta \cdot \alpha} + 1\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   4. associate-+l+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   5. +-commutative 83.2%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   6. associate-+l+ 83.2%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   7. *-commutative 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   8. associate-*r* 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   9. associate-+r+ 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   10. +-commutative 83.2%

       \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   11. associate-/l/ 93.2%

       \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   12. clear-num 93.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   13. inv-pow 93.1%

       \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

6. Applied egg-rr 93.2%

   \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 93.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. associate-/l* 94.0%

      \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   3. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   4. +-commutative 94.0%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   5. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   6. associate-+r+ 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   7. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   8. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \color{blue}{\left(\beta + \alpha\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   9. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

   10. fma-undefine 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

   11. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   12. *-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   13. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   14. associate-+r+ 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   15. distribute-lft1-in 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   16. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   18. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

   19. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

8. Simplified 94.0%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

9. Taylor expanded in beta around 0 69.5%

   \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]

10. Taylor expanded in alpha around 0 45.2%

    \[\leadsto \frac{1}{\color{blue}{6 \cdot \left(2 + \beta\right)}} \]

11. Final simplification 45.2%

    \[\leadsto \frac{1}{\left(\beta + 2\right) \cdot 6} \]
12. Add Preprocessing

Alternative 20: 47.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\beta + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ beta 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0 / (beta + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (beta + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(beta + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (beta + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Simplified 83.2%

   \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   2. fma-undefine 83.2%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \color{blue}{\left(\alpha \cdot \beta + 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   3. *-commutative 83.2%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \left(\color{blue}{\beta \cdot \alpha} + 1\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   4. associate-+l+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   5. +-commutative 83.2%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   6. associate-+l+ 83.2%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   7. *-commutative 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   8. associate-*r* 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   9. associate-+r+ 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   10. +-commutative 83.2%

       \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   11. associate-/l/ 93.2%

       \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   12. clear-num 93.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   13. inv-pow 93.1%

       \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

6. Applied egg-rr 93.2%

   \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 93.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. associate-/l* 94.0%

      \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   3. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   4. +-commutative 94.0%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   5. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   6. associate-+r+ 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   7. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   8. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \color{blue}{\left(\beta + \alpha\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   9. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

   10. fma-undefine 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

   11. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   12. *-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   13. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   14. associate-+r+ 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   15. distribute-lft1-in 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   16. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   18. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

   19. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

8. Simplified 94.0%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

9. Taylor expanded in beta around 0 69.5%

   \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]

10. Taylor expanded in alpha around 0 44.9%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
11. Step-by-step derivation

    1. +-commutative 44.9%

       \[\leadsto \frac{0.16666666666666666}{\color{blue}{\beta + 2}} \]

12. Simplified 44.9%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

13. Final simplification 44.9%

    \[\leadsto \frac{0.16666666666666666}{\beta + 2} \]
14. Add Preprocessing

Alternative 21: 46.1% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 94.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Simplified 83.2%

   \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   2. fma-undefine 83.2%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \color{blue}{\left(\alpha \cdot \beta + 1\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   3. *-commutative 83.2%

      \[\leadsto \frac{\left(\alpha + \beta\right) + \left(\color{blue}{\beta \cdot \alpha} + 1\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   4. associate-+l+ 83.2%

      \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   5. +-commutative 83.2%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   6. associate-+l+ 83.2%

      \[\leadsto \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   7. *-commutative 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \]

   8. associate-*r* 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\color{blue}{\left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   9. associate-+r+ 83.2%

      \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\color{blue}{\left(\left(\beta + \alpha\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   10. +-commutative 83.2%

       \[\leadsto \frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   11. associate-/l/ 93.2%

       \[\leadsto \color{blue}{\frac{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   12. clear-num 93.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   13. inv-pow 93.1%

       \[\leadsto \color{blue}{{\left(\frac{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}{\frac{1 + \left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]

6. Applied egg-rr 93.2%

   \[\leadsto \color{blue}{{\left(\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 93.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   2. associate-/l* 94.0%

      \[\leadsto \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}}} \]

   3. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\alpha + \color{blue}{\left(2 + \beta\right)}\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   4. +-commutative 94.0%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + \beta\right) + \alpha\right)} \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   5. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\beta + 2\right)} + \alpha\right) \cdot \frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   6. associate-+r+ 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{\left(\alpha + \beta\right) + 3}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   7. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{\color{blue}{3 + \left(\alpha + \beta\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   8. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \color{blue}{\left(\beta + \alpha\right)}}{\frac{1 + \mathsf{fma}\left(\alpha + 1, \beta, \alpha\right)}{\alpha + \left(\beta + 2\right)}}} \]

   9. +-commutative 94.0%

      \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\mathsf{fma}\left(\alpha + 1, \beta, \alpha\right) + 1}}{\alpha + \left(\beta + 2\right)}}} \]

   10. fma-undefine 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\left(\alpha + 1\right) \cdot \beta + \alpha\right)} + 1}{\alpha + \left(\beta + 2\right)}}} \]

   11. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\left(1 + \alpha\right)} \cdot \beta + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   12. *-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\color{blue}{\beta \cdot \left(1 + \alpha\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   13. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(\beta \cdot \color{blue}{\left(\alpha + 1\right)} + \alpha\right) + 1}{\alpha + \left(\beta + 2\right)}}} \]

   14. associate-+r+ 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\beta \cdot \left(\alpha + 1\right) + \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   15. distribute-lft1-in 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   16. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}}} \]

   17. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)}}} \]

   18. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}} \]

   19. +-commutative 94.0%

       \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}} \]

8. Simplified 94.0%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \frac{3 + \left(\beta + \alpha\right)}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \]

9. Taylor expanded in beta around 0 69.5%

   \[\leadsto \frac{1}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \color{blue}{\frac{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}{1 + \alpha}}} \]

10. Taylor expanded in alpha around 0 44.9%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]
11. Step-by-step derivation

    1. +-commutative 44.9%

       \[\leadsto \frac{0.16666666666666666}{\color{blue}{\beta + 2}} \]

12. Simplified 44.9%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{\beta + 2}} \]

13. Taylor expanded in beta around 0 43.7%

    \[\leadsto \color{blue}{0.08333333333333333} \]

14. Final simplification 43.7%

    \[\leadsto 0.08333333333333333 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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)))